11institutetext: Observatoire de Genève, Université de Genève, Chemin Pegasi,
1290 Versoix, Switzerland
11email<EMAIL_ADDRESS>22institutetext: School of Earth and Space
Exploration, Arizona State University, 781 Terrace Mall, Tempe, AZ 85287, USA
33institutetext: CNRS, IRAP, 14 Avenue E. Belin, 31400 Toulouse, France
44institutetext: National Radio Astronomy Observatory, P.O. Box O, Socorro, NM
87801, USA 55institutetext: Department of Physics and Astronomy, University of
Manitoba, Winnipeg, Canada 66institutetext: Department of Astronomy,
University of Massachusetts Amherst, Amherst, MA 01002, United States
77institutetext: Department of Astronomy, The University of Texas at Austin,
2515 Speedway, Stop C1400, Austin, TX 78712-1205, USA 88institutetext:
Department of Astronomy, Williams College, Williamstown, MA 01267, United
States 99institutetext: Space Telescope Science Institute, 3700 San Martin
Drive, Baltimore, MD, 21218, USA 1010institutetext: VDI/VDE Innovation +
Technik GmbH 1111institutetext: Steward Observatory, University of Arizona,
933 N. Cherry Avenue, Tucson, AZ 85721, USA 1212institutetext: Department of
Astronomy, University of Michigan, 1085 S. University, Ann Arbor, MI 48109,
USA 1313institutetext: Kapteyn Astronomical Institute, University of
Groningen, P.O. Box 800, 9700 AV Groningen, The Netherlands 1414institutetext:
Stockholm University, Department of Astronomy and Oskar Klein Centre for
Cosmoparticle Physics, AlbaNova University Centre, SE-10691, Stockholm, Sweden
1515institutetext: INAF- Osservatorio, Astronomico di Roma, Via di FRascati,
33 00078, Monte Porzio Catone (Italy) 1616institutetext: Astronomy department,
University of Virginia, P.O. Box 400325, Charlottesville, VA 22904-4325, USA
1717institutetext: ARAID Foundation. Centro de Estudios de Física del Cosmos
de Aragón (CEFCA), Unidad Asociada al CSIC, Plaza San Juan 1, E–44001 Teruel,
Spain 1818institutetext: Departamento de Astronomía, Universidad de La Serena,
Av. Juan Cisternas 1200 Norte, La Serena 1720236, Chile 1919institutetext:
Department of Astronomy & Astrophysics, The Pennsylvania State University,
University Park, PA 16802, USA 2020institutetext: Institute for Computational
& Data Sciences, The Pennsylvania State University, University Park, PA 16802,
USA 2121institutetext: Institute for Gravitation and the Cosmos, The
Pennsylvania State University, University Park, PA 16802, USA
2222institutetext: Center for Astrophysical Sciences, Department of Physics &
Astronomy, Johns Hopkins University, Baltimore, MD 21218, USA
2323institutetext: International Space Science Institute (ISSI), Hallerstrasse
6, CH-3012 Bern, Switzerland

# The Low-redshift Lyman Continuum Survey:

Radio continuum properties of low-$z$ Lyman continuum emitters

Omkar Bait , O. Bait is a corresponding author11 Sanchayeeta Borthakur 22
Daniel Schaerer 1133 Emmanuel Momjian 44 Biny Sebastian 55 Alberto Saldana-
Lopez 11 Sophia R. Flury 66 John Chisholm 77 Rui Marques-Chaves 11 Anne E.
Jaskot 88 Harry C. Ferguson 99 Gabor Worseck 1010 Zhiyuan Ji 1111 Lena
Komarova 1212 Maxime Trebitsch 1313 Matthew J. Hayes 1414 Laura Pentericci
1515 Goran Ostlin 1414 Trinh Thuan 1616 Ricardo O. Amorín 1717 18 18 Bingjie
Wang 1919 20 20 21 21 Xinfeng Xu 2222 Mark T. Sargent 2323

(Received ; accepted )

###### Abstract

Context.  The sources that leak Lyman-continuum (LyC) photons and lead to the
reionisation of the universe are intensely studied using multiple observing
facilities. Recently, the Low-redshift LyC Survey (LzLCS) has found the first
large sample of LyC emitting galaxies at low redshift ($z\sim 0.3$) with the
Hubble Space Telescope /Cosmic Origins Spectrograph. The LzLCS sample contains
a robust estimate of the LyC escape fraction
($f_{\mathrm{esc}}^{\mathrm{LyC}}$) for 66 galaxies spanning a wide range of
$f_{\mathrm{esc}}^{\mathrm{LyC}}$.

Aims.  Here we aim to study the radio continuum (RC) properties of LzLCS
sources and their dependence on $f_{\mathrm{esc}}^{\mathrm{LyC}}$. RC can
provide unique insights into the role of supernova feedback, cosmic-rays (CRs)
and magnetic fields from its non-thermal emission component. RC is also a
dust-free tracer of the star-formation rate (SFR) in galaxies.

Methods.  In this study, we present Karl G. Jansky Very Large Array (VLA) RC
observations of the LzLCS sources at gigahertz (GHz) frequencies. We performed
VLA C (4-8 GHz) and S (2-4 GHz) band observations for a sample of 53 LzLCS
sources. We also observed a sub-sample of 17 LzLCS sources in the L (1-2 GHz)
band. We detected RC from both C- and S-bands in 24 sources for which we can
estimate their radio spectral index across 3-6 GHz, denoted as
$\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$. We also use the RC luminosity to
estimate their SFRs.

Results. The radio spectral index of LzLCS sources spans a wide range from
being flat ( $\geq-0.1$) to very steep ($\leq-1.0$) compared to that of normal
star-forming galaxies ($\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$$=-0.64$). We
find that the strongest leakers in our sample show flat
$\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$, weak leakers have
$\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$ close to normal star-forming
galaxies, and non-leakers are characterized by steep
$\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$. We argue that a combination of young
ages, free-free absorption, and a flat cosmic-ray energy spectrum can
altogether lead to a flat $\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$ for strong
leakers. Non-leakers are characterized by steep spectra which can arise due to
break/cutoff at high frequencies. Such a cutoff in the spectrum can arise in a
single injection model of CRs characteristic of galaxies which have recently
stopped star-formation. Alternatively, CR escape due to advection/diffusion
can also steepen their spectra. The radio-based SFRs of LzLCS sources show a
large scatter around the standard RC-SFR calibration.

Conclusions. In this study we, for the first time, find a relation between
$\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$ and
$f_{\mathrm{esc}}^{\mathrm{LyC}}$. This hints at the interesting role of
supernovae feedback, CRs, and magnetic fields in facilitating the escape
(and/or the lack) of LyC photons.

###### Key Words.:

galaxies: starburst – Radio continuum: galaxies – Radio continuum: ISM

##  1 Introduction

In order to understand how and when the Universe underwent reionisation it is
crucial to know the nature of the sources that provide such ionising UV
photons. Although quasars are very bright in the UV, their number density at
$z\geq 4$ is likely too low to contribute significantly to the reionising
budget of the Universe (see discussions in Kulkarni et al., 2019; Giallongo et
al., 2015; Grazian et al., 2020). However, at high redshift, low-mass star-
forming galaxies dominate the luminosity (mass) functions (e.g., Ouchi et al.,
2009; Bouwens et al., 2015; Grazian et al., 2015; Song et al., 2016) which can
be significant contributors to reionising the Universe (Robertson et al.,
2010; Ishigaki et al., 2018; Finkelstein et al., 2019). And bright galaxies
could also contribute to reionisation, as discussed by several authors (see
e.g., Sharma et al., 2016; Naidu et al., 2020; Marques-Chaves et al., 2021,
2022a).

The escape of Lyman continuum (LyC) photons from galaxies before the Universe
was completely reionised cannot be directly studied since the LyC photons will
be absorbed by the inter-galactic medium (IGM) (Inoue et al., 2014). Thus, we
have to rely on local (low redshift) analogues of high-redshift galaxies to
measure the escape of LyC photons where the IGM is mostly transparent to
ionizing photons. Moreover, low redshift galaxies are easier to study in
detail. Typically these local analogues ($z\sim 0.3$) are extremely young
starbursts with low dust attenuation (characterized by the slope of the non-
ionizing UV continumm, $\beta\leq-2$ given by
$f_{\lambda}\propto\lambda^{\beta}$) e.g., Chisholm et al. (2022), with high
star-formation rate density ($\Sigma_{\mathrm{SFR}}$ $\geq 0.1$ M⊙yr-1kpc-2)
and high ionization ratio ([OIII]/[OII]; O32$\geq 3$) and have several other
extreme emission line properties (e.g., Cardamone et al., 2009; Yang et al.,
2017; Izotov et al., 2011, 2021a). These galaxies can efficiently produce
ionising photons (Schaerer et al., 2016) and also leak a large amount of them
(Izotov et al., 2016b, c, 2018a, 2018b, 2021b), thus their counterparts at
high-$z$ can significantly contribute to reionisation. Recent James Webb Space
Telescope (JWST) observations confirmed that these are excellent local
analogues of $z>6$ galaxies, thus during the epoch of reionisation, since they
show several similar extreme emission line properties (e.g., Schaerer et al.,
2022; Rhoads et al., 2023; Matthee et al., 2022; Cameron et al., 2023; Mascia
et al., 2023).

A systematic study of low redshift LyC emitters (LCEs, often also referred to
as LyC leakers) was lacking until recently. The high success rate in finding
LCEs in low-$z$ analogues has motivated the Low-$z$ Lyman Continuum Survey
(Flury et al., 2022a, LzLCS, PI: Jaskot HST Project ID: 15626,). LzLCS is a
systematic survey of 66 star-forming galaxies at $z\sim 0.3$ using Hubble
Space Telescope (HST)/Cosmic Origins Spectrograph (COS) to measure their LyC.
They are selected from the Sloan Digital Sky Survey (SDSS) and Galaxy
Evolution Explorer (GALEX), with a range of O32, star-formation rate densities
($\Sigma_{\mathrm{SFR}}$) and UV-slopes $\beta$. The LzLCS has found strong
($>2\sigma$) LyC detection in 35 galaxies with a range of LyC escape fractions
($f_{\mathrm{esc}}^{\mathrm{LyC}}$ up to $\sim$ 50%), and has been combined
with previous LyC measurements to provide a sample of 89 low-$z$ galaxies with
LyC measurements. Such a sample can then be used to study the dependence of
various UV/optical-based physical properties on
$f_{\mathrm{esc}}^{\mathrm{LyC}}$ (Flury et al., 2022b; Saldana-Lopez et al.,
2022; Marques-Chaves et al., 2022b; Chisholm et al., 2022; Wang et al., 2021;
Xu et al., 2023).

The exact physical mechanisms which lead to the leakage of LyC photons are
still a matter of debate. LyC photons can either escape through low Hi column
density holes carved in the interstellar medium (ISM) (Gazagnes et al., 2018;
Saldana-Lopez et al., 2022) or from a density-bounded nebula (e.g., Nakajima &
Ouchi, 2014; Zackrisson et al., 2013; Bremer & Dayal, 2023). Simulations show
that strong supernova (SN) driven feedback can play an important role in the
escape of LyC photons (Kimm & Cen, 2014; Paardekooper et al., 2015; Kimm et
al., 2017; Trebitsch et al., 2017; Ma et al., 2020; Barrow et al., 2020).
Kakiichi & Gronke (2021) also shows the effect of turbulent ISM on the escape
of LyC photons. And recently (Katz et al., 2023) showed that escape of LyC
photons can occurs from different modes of star-formation. Ma et al. (2020) in
their zoom-in cosmological simulations find that most LCEs are associated with
a feedback-driven kiloparsec-scale superbubble and host very young stellar
populations ($\la 10$ Myr). Observationally, this is seen for example in the
Sunburst arc where the regions which are leaking show a very young stellar
population and outflow signatures as opposed to the non-leaking regions
(Mainali et al., 2022; Kim et al., 2023). Simulations have also highlighted
the rapid fluctuations (10 Myr timescales) due to the stochastic/bursty nature
of star formation (e.g., Ma et al., 2015; Trebitsch et al., 2017; Kimm et al.,
2017; Barrow et al., 2020; Endsley et al., 2023). All these effects complicate
the physical understanding and predictability of
$f_{\mathrm{esc}}^{\mathrm{LyC}}$ using various diagnostics.

Radio observations provide a complimentary view into these galaxies. For
example, the radio continuum (RC) is a dust-free tracer of the total star-
formation rate (SFR) in galaxies (Yun & Carilli, 2002; Murphy et al., 2011).
For normal galaxies, the RC at 1.4 GHz and other mid-GHz frequencies (3, 6 and
10 GHz) has been extensively used to measure the SFR of galaxies both nearby
and at high redshifts (e.g., Kennicutt & Evans, 2012; Tabatabaei et al.,
2017). Modelling the radio-spectral energy distribution (radio-SED),
particularly across several widely separated frequencies allows us to study
several unique physical parameters (Tabatabaei et al., 2017), for example, the
supernova rate, the role of cosmic-rays (CRs), emission measure based on the
free-free absorption optical depth (Condon, 1992; Hunt et al., 2004), and the
equipartition magnetic field (Beck & Krause, 2005; Beck, 2015).

RC emission at gigahertz frequencies in normal galaxies arises due to a
combination of non-thermal synchrotron emission, which approximately follows a
power-law with a slope, $\alpha_{\mathrm{nth}}$ $\sim-0.8$, and a thermal
free-free component with a power law slope of $-0.1$ (Condon et al., 1991). In
more complex scenarios the radio-SED shows a deviation from a simple power law
behaviour. For example, in the case of dense starbursts in Ultra Luminous
Infrared Galaxies (ULIRGS), blue-compact dwarfs (BCDs) and metal-poor dwarf
galaxies, it is observed that the radio-SED shows a turnover below
$\sim\leavevmode\nobreak\ 1-2$ GHz (e.g., Condon et al., 1991; Hunt et al.,
2004). Such a turnover in dense starbursts can arise from free-free absorption
(FFA) due to large optical depths. Moreover, the RC spectrum can also show
multiple FFA components (e.g., Clemens et al., 2010; Galvin et al., 2018). In
addition at mid-high frequencies, they can show a much steeper spectrum than
the canonical slope of -0.7, due to a steeper non-thermal spectral index
(Galvin et al., 2018). Deviations in the non-thermal spectral index
($\alpha_{\mathrm{nth}}$) are also observed in local BCDs, showing both
steeper and flatter $\alpha_{\mathrm{nth}}$ than normal galaxies, possibly due
to the inhomogeneous nature of star-formation in these galaxies (e.g., Klein
et al., 1984, 1991; Deeg et al., 1993). Steep $\alpha_{\mathrm{nth}}$ values
are associated with a general lack of CR confinement (Skillman & Klein, 1988;
Klein et al., 1991). The exact physical mechanisms for this steepening are not
well known, but it can be due to synchrotron ageing particularly in a strong
magnetic field, inverse-Compton (IC) losses or even the escape of CRs (see
Klein et al., 1991; Ramya et al., 2011, for a discussion). Such deviations
have systematic effects on the radio-SED and can be used to distinguish
different physical processes (see e.g., Klein et al., 2018; Clemens et al.,
2010; Galvin et al., 2018). Overall the radio spectral index can also be used
as a proxy for the age of starburst (Cannon & Skillman, 2004; Hirashita &
Hunt, 2006). In young, metal-poor and dense starbursts (e.g., SBS 0335-052;
Hunt et al., 2004) the non-thermal emission is associated with an ensemble of
compact SN remnants (SNRs) which are expanding in a dense interstellar medium
(e.g., Hunt et al., 2004; Hirashita & Hunt, 2006) instead of diffuse CRs in
normal galaxies.

Of particular interest is the non-thermal luminosity which along with the non-
thermal spectral index can be used to estimate the Type II SN rate (Condon,
1992). This can help us in understanding the role of SN feedback, and thus
also indirectly the contribution of SN to the escape of ionizing photons. The
radio spectral index and the deviations between radio and other UV/optical/IR
tracers are also used to understand changes in the recent star-formation
history (e.g., Cannon & Skillman, 2004; Bressan et al., 2002; Hirashita &
Hunt, 2006; Arango-Toro et al., 2023). There is no systematic RC study of LCEs
to date. RC studies of local LCEs on Haro 11 (Hayes et al., 2007) and
Tol1247-232 (Puschnig et al., 2017) have found a lower radio-based SFR
compared to the SFR from other tracers. RC studies of sources most closely
resembling LCEs e.g., Green-Peas (GPs) and Blueberries (BBs) also have
suppressed radio-based SFR compared to other SFR indicators (Chakraborti et
al., 2012; Sebastian & Bait, 2019). It is speculated that such a suppression
could be because galaxies with young stellar populations do not have any SN
activity yet and thus have lower non-thermal emission component than normally
expected (Hayes et al., 2007; Chakraborti et al., 2012; Sebastian & Bait,
2019). Moreover, loss of CRs either via diffusion or outflows might also be
playing a role in causing the deficit (Sebastian & Bait, 2019). Such a deficit
has also been observed in local analogues of Lyman-break galaxies (LBGs)
(Greis et al., 2017) and for $z\sim 3$ LBGs based on a stacking analysis
(Carilli et al., 2008). The exact causes for this deficit are not yet
completely investigated due to their unknown radio-SED. Several of the factors
mentioned above for powerful starbursts are also valid for GPs and BBs which
can lead to these variations.

In this study, for the very first time, we present RC observations of low-$z$
star-forming galaxies with LyC measurements, which allows us to examine if and
how their RC emission relates to the escape of ionizing photons. For instance,
the radio thermal fraction and spectral index depend on the past SN activity
and star-formation history (SFH) of galaxies. Importantly, the radio spectral
index can be used as a proxy for the age of the starburst (Cannon & Skillman,
2004; Hirashita & Hunt, 2006). Dependence of these radio properties on
$f_{\mathrm{esc}}^{\mathrm{LyC}}$ can tell us for example about the role of
SN-driven feedback and SFH of galaxies on the escape of LyC photons. To do so,
we target $z\sim 0.3-0.4$ galaxies from the LzLCS sample. Our observations are
conducted using the NRAO111The National Radio Astronomy Observatory is a
facility of the National Science Foundation operated under cooperative
agreement by Associated Universities, Inc. Karl G. Jansky Very Large Array
(VLA) for a sample of 53 LzLCS sources at 6 (C-band), 3 (S-band) and also for
a subsample at 1.5 (L-band) GHz. Our paper is organized as follows. In Section
2 we describe our observation setup, data analysis and detection and non-
detection statistics. In Section 3, we present the radio spectral index
(across 3-6 GHz; $\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$) and thermal
fraction distribution, and radio-SED for our sample of detections. Our main
result is described in Section 3.4 where we show a dependence between
$\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$and $f_{\mathrm{esc}}^{\mathrm{LyC}}$.
In Section 4, we discuss the various physical reasons that can give rise to
such a relation. In Section 5, we summarize our main results and conclude.

Throughout this paper we assume $H_{0}=70$ km s-1 Mpc-1, $\Omega_{m}=0.3$, and
$\Omega_{\Lambda}=0.7$, which is same as that used in Flury et al. (2022a).

##  2 Data

In this section, we describe our sample selection and VLA observations at
multiple observing bands, namely the C (4-8 GHz), S (2-4 GHz) and L (1-2 GHz)
with the B-configuration (PI: Borthakur; Project Code: 21B-111). We describe
our calibration and imaging strategy.

###  2.1 Sample Selection

Our current sample consists of 53 LzLCS sources which we observed with the
VLA. This sample is a subset of the complete 66 galaxies from LzLCS. The LzLCS
sample spans a wide range in $f_{\mathrm{esc}}^{\mathrm{LyC}}$ and can hence
be used to do a statistical study of LCEs and its dependence on other physical
properties (Flury et al., 2022b; Saldana-Lopez et al., 2022; Marques-Chaves et
al., 2022b; Xu et al., 2023). Our 53 LzLCS sources with VLA observations were
selected such that their expected flux densities at L-band is above 40 $\mu$Jy
at 5-$\sigma$. This corresponds to a SFR of 6.6 $\mathrm{M}_{\odot}$/yr for
the median redshift ($z\sim 0.3$) of our sample based on the standard radio-
SFR relation (Murphy et al., 2011).

###  2.2 Observation setup

We used the default VLA setup for our C-, S- and L- band continuum
observations. For the C-band (4-8 GHz) we used a 3-bit sampler with a total
bandwdith of 4 GHz and for the S-band (2-4 GHz) and L-band (1-2 GHz) we used
the 8-bit sampler with a total bandwidth of 2 and 1 GHz respectively. For the
C-band setup, we split the entire bandwidth into two basebands each having 16
$\times$ 128 MHz subbands. Each subband was further split into 64 channels.
Similarly for the S-band setup, we split the entire bandwidth into two
basebands each having 8 $\times$ 128 MHz subbands where each subband was
further split into 64 channels. And for the L-band setup, we split the entire
bandwidth into two basebands each having 8 $\times$ 64 MHz subbands where each
subband was further split into 64 channels. All our observations were
conducted in the B-configuration with a few in the BtoBnA, BnA and BnAtoA move
time, depending on the scheduling constraints. We could thus typically achieve
an angular resolution of 1.6″, 3.2″ and 6.9″ for C-, S- and L-bands
respectively.

We followed the standard observing strategy used for continuum observations.
We typically spend a minute on the initial setup, and then 3-5 minutes on the
flux density scale calibrator. We then observe the phase calibrator for a
minute and the target for 5-6 minutes. We then cycle between the phase
calibrator and the target. We have a total on-source time of 22 minutes for C-
and S-band observations, and a total on-source time of 75 minutes for the
L-band observations. The C- and S-band observations of a given target source
were combined in a single $\sim$ 1 hr long scheduling block (SB), while their
L-band observations were in separate SB with a length of $\sim$1 hr and 40
min. Table 4 and 5 summarizes the S-, C- and L-band observing details of our
sample. In total we could target 53 sources in both the S- and C-bands, and a
subset of 17 in the L-band.

###  2.3 Data Analysis: Calibration and Imaging

We used the Common Astronomy Software Applications (CASA) data processing
software (McMullin et al., 2007; THE CASA TEAM et al., 2022) for all our data
analysis. Each SB containing the multi-band C- and S-band data was initially
flagged and calibrated using the VLA calibration pipeline v6.2.1. The
calibrated flux density scale and phase calibrators for each SB were manually
inspected for any leftover RFI and poorly performing antennas. These were then
flagged and the datasets were re-run using the latest CASA pipeline v6.4.1.
For some datasets, several rounds of the pipeline were run, after having
performed flagging in each round.

The final calibrated target data were then extracted and any leftover RFI in
some spectral windows was flagged using the task rflag. We then imaged these
data using CASA task tclean, separately by splitting the target in C- and
S-bands. We use the multi-term multi-frequency synthesis (mtmfs) algorithm
(Rau & Cornwell, 2011) with nterms=2, and a robust=0.5. We then manually make
elliptical regions around each of our target sources using CASA task imview.
We then estimated the flux densities of these sources using 2D Gaussian fit
using the task imfit. We use this same region to estimate the noise in the
images from a emission free area. The estimated flux densities and noise
($\sigma$) for our sources in C-, S- and L-bands are presented in Table 1. For
sources with signal-to-noise ratio below 3, we quote the 3$\sigma$ upper
limits in the flux density. In Table 2 we present a summary of the number of
detections and non-detections in different bands .

Table 1: Radio Properties of our LzLCS sample.

Object | Flux density at $3$ GHz | Flux density at 6 GHz | Flux density at 1.5 GHz | $\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$
---|---|---|---|---
| ($\mu$Jy) | ($\mu$Jy) | ($\mu$Jy) |
J004743+015440 | $<28.5$ | $24.0\pm 5.5$ | $<90.0$ | $<-0.25$
J011309+000223a | $<31.2$ | $<28.5.0$ | $<63.0$ | –
J012910+145935 | $<45.0$ | $75.6\pm 7.0$ | $<63.0$ | $<0.75$
J072326+414608 | $87.2\pm 8.1$ | $<16.5$ | $<42.0$ | $>-2.4$
J081112+414146 | $<44.4$ | $<19.5$ | $<42.0$ | –
J081409+211459 | $142.0\pm 8.8$ | $114.3\pm 5.2$ | $300.0\pm 8.0$ | $-0.313\pm-0.024$
J082652+182052 | $69.4\pm 8.9$ | $37.0\pm 6.6$ | $<27.0$ | $-0.907\pm-0.199$
J083440+480541 | $68.2\pm 7.8$ | $69.1\pm 5.1$ | $143.0\pm 15.0$ | $0.019\pm 0.003$
J090918+392925 | $<25.5$ | $<17.4$ | $<45.0$ | –
J091113+183108 | $69.2\pm 8.5$ | $52.7\pm 5.0$ | $<90.0$ | $-0.393\pm-0.061$
J091207+523960a | $<68.1$ | $<22800$ | $<123.0$ | –
J091208+505009 | $<26.1$ | $40.0\pm 5.7$ | $<45.0$ | $<0.62$
J091703+315221 | $55.6\pm 8.2$ | $47.5\pm 5.8$ | $80.0\pm 20.0$ | $-0.227\pm-0.043$
J092552+395714 | $<25.8$ | $<16.5$ | $<33.0$ | –
J094001+593244 | $63.0\pm 10.4$ | $26.0\pm 4.9$ | – | $-1.277\pm-0.32$
J095236+405249b | $392.3\pm 7.5$ | $50.9\pm 5.4$ | $<36.0$ | $-2.946\pm-0.318$
J095700+235709 | $753.3\pm 10.0$ | $224.5\pm 5.6$ | $300.0\pm 20.0$ | $-1.747\pm-0.049$
J095838+202508 | $<25.2$ | $<16.5$ | $<30.0$ | –
J101401+523251 | $<22.5$ | $<16.2$ | – | –
J102615+633308 | $<25.2$ | $<15.6$ | – | –
J103344+635317 | $73.8\pm 8.0$ | $80.7\pm 6.0$ | – | $0.129\pm 0.017$
J103816+452718 | $132.4\pm 8.9$ | $40.4\pm 5.8$ | – | $-1.712\pm-0.271$
J105117+474357 | $<27.0$ | $<16.8$ | – | –
J105331+523753 | $158.7\pm 8.2$ | $79.8\pm 5.6$ | – | $-0.992\pm-0.086$
J110452+475204 | $<23.7$ | $<16.2$ | – | –
J112224+412052 | $<25.5$ | $<17.1$ | – | –
J112848+524509 | $<23.1$ | $<16.2$ | – | –
J112933+493525 | $<22.2$ | $<18.0$ | – | –
J113304+651341 | $<24.0$ | $27.9\pm 5.8$ | – | $<0.22$
J115855+312559 | $136.7\pm 7.9$ | $72.9\pm 5.6$ | – | $-0.907\pm-0.087$
J115959+382422 | $<23.7$ | $<17.1$ | – | –
J120934+305326 | $120.7\pm 7.8$ | $70.5\pm 5.5$ | – | $-0.776\pm-0.079$
J121915+453930 | $<25.5$ | $<19.2$ | – | –
J123519+063556 | $<28.5$ | $<19.5$ | – | –
J124423+021540 | $225.6\pm 9.2$ | $145.7\pm 5.9$ | – | $-0.631\pm-0.036$
J124619+444902 | $61.0\pm 12.6$ | $41.5\pm 5.0$ | – | $-0.556\pm-0.133$
J124835+123403 | $126.0\pm 7.7$ | $76.0\pm 6.0$ | – | $-0.729\pm-0.073$
J124911+464535 | $57.0\pm 7.9$ | $<16.8$ | – | $>-1.76$
J125718+410221b | $82.2\pm 9.7$ | $26.6\pm 5.5$ | – | $-1.628\pm-0.388$
J130128+510451 | $488.0\pm 8.0$ | $<16.5$ | – | $>-4.89$
J131037+214817 | $<24.3$ | $41.3\pm 5.6$ | – | $<0.77$
J131419+104739 | $125.0\pm 8.3$ | $37.7\pm 6.0$ | – | $-1.2\pm-0.298$
J131904+510309 | $122.5\pm 8.7$ | $30.4\pm 5.0$ | – | $-1.57\pm-0.36$
J132633+421824 | $95.0\pm 8.4$ | $<17.1$ | – | $>-2.47$
J132937+573315 | $132.1\pm 8.2$ | $29.2\pm 5.6$ | – | $-2.178\pm-0.439$
J134559+112848 | $249.6\pm 7.7$ | $71.4\pm 5.7$ | – | $-1.806\pm-0.155$
J141013+434435b | $233.0\pm 8.8$ | $99.6\pm 5.4$ | – | $-1.226\pm-0.081$
J144010+461937 | $158.9\pm 8.3$ | $96.7\pm 5.7$ | – | $-0.717\pm-0.056$
J151707+370512 | $<30.0$ | $30.7\pm 6.4$ | – | $<0.03$
J155945+403325 | $74.8\pm 8.3$ | $87.2\pm 5.3$ | – | $0.221\pm 0.028$
J160437+081959 | $113.6\pm 9.4$ | $<18.0$ | – | $>-2.66$
J164607+313054 | $<26.7$ | $<19.8$ | – | –
J164849+495751 | $<39.9$ | $<38.4$ | – | –

222 For non-detections in the different bands, we present the $3\sigma$ upper
limits on the flux density. For estimating the upper/lower limits on
$\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$, we use the corresponding $3\sigma$
upper limits on the flux density.
a Here the noise is high due to a very strong source in the field.
b For these sources the flux density measurement is uncertain due to artefacts
from a strong source in the field.

Table 2: Detection and Non-detection statistics for our sample of 53 LzLCS
sources. $\begin{array}[]{p{0.8\linewidth}l}\hline\cr\vskip 3.0pt plus 1.0pt
minus 1.0pt\cr Detection/Non-detection Type&Number\\\ \vskip 3.0pt plus 1.0pt
minus 1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt minus 1.0pt\cr Both C- \&
S-band detections&24\\\ Both C- \& S-band non-detections&18\\\ C-band only
non-detections&5\\\ S-band only non-detections&6\\\ \vskip 3.0pt plus 1.0pt
minus 1.0pt\cr\hline\cr\end{array}$

###  2.4 VLA images

Figure 1:  The VLA images of two sources in our sample, J$124423+021540$ (top
panels) and J$095700+235709$ (bottom panels) show in white contours with the
SDSS r-band image (Blanton et al., 2017) in the background. On the left
(right) panels we show the VLA S-band (C-band) image contours. The lowest
contours in each panels corresponds to 3$\sigma$ level and they rise in powers
of $\sqrt{2}$. The corresponding $\sigma$ values are presented in Table 1.

In Fig. 1 upper panel (lower panel), we overlay the S- and C-band radio
contours on SDSS r-band images in the background for J124423+021540
(J095700+235709). In Appendix B, we show the C- and S-band VLA images for the
rest of the detection sample. Notice that except for J095700+235709, shown in
Fig. 1, all the sources in our sample are unresolved in both radio and optical
wavelengths.

##  3 Results

###  3.1 Radio-SED at mid GHz frequencies

Figure 2: Shows the typical radio-SED with a mixture of thermal and non-
thermal emission (Eq. 1) in blue line. The corresponding thermal and non-
thermal emission components are shown in blue dashed and dotted lines. We also
show the effect of FFA (Eq. 2) with a turnover at 1 GHz in orange. And the
cutoff and curved model (from Klein et al., 2018) with a high-frequency break
at 6 GHz in green and red respectively. The grey shaded area shows the region
of the SED explored in this work.

Figure 3: Observed radio continuum SEDs at three frequencies, 1.5, 3, and 6
GHz for three sources in our sample. The error bars represent $1\sigma$ errors
on the flux measurement. The gray line in each panel is an overlay (not a fit)
for a simple radio-SED (Eq. 1) normalized to the 3 GHz observed flux density
with $f_{\mathrm{th}}=0.1$ at 1 GHz and $\alpha_{\mathrm{nth}}=-0.8$ commonly
seen in star-forming galaxies. The left panel shows an example of a radio-SED
which is close to the standard radio-SED (Eq. 1). The middle panel shows an
example of a spectrum that needs a turnover below 3GHz. Above 3 GHz it has a
relatively steeper spectrum, thus appearing like a Gigahertz Peak Spectrum
(GPS) source. The right panel shows the case of a broken SED.

The RC in the simplest case is modeled by a single power law given by:333In
what follows we use this sign convention to define the spectral index. Note
that some references use the opposite sign convention.$S_{\nu}=A\nu^{\alpha},$
where $\alpha$ is the power-law index, also termed the total spectral index,
$\nu$ is the frequency, and $A$ is a proportionality constant. At frequencies
below 30 GHz, the radio-SED for normal star-forming galaxies can be modeled as
a combination of i) free-free thermal emission from star-forming regions
having a relatively flat spectral index of $-0.1$ and ii) non-thermal emission
having a steeper spectral index ($\alpha_{\mathrm{nth}}$) arising due to
synchrotron emission from relativistic electrons (see e.g., Condon, 1992).
Thus, the RC SED in such a case can be written with respect to a reference
frequency $\nu_{\circ}$ following Tabatabaei et al. (2017), as:

$S_{\nu}=S_{\nu}^{\mathrm{th}}+S_{\nu}^{\mathrm{nth}}=A_{1}\left(\frac{\nu}{\nu_{\circ}}\right)^{-0.1}+A_{2}\left(\frac{\nu}{\nu_{\circ}}\right)^{\alpha_{\mathrm{nth}}}.$
(1)

We can then define the thermal fraction at the reference frequency
$f_{\mathrm{th}}{}=S_{\nu_{\circ}}^{\mathrm{th}}/S_{\nu_{\circ}}$.

Under the assumption that the thermal emission is optically thin and no escape
of LyC photons, $S_{\nu}^{\mathrm{th}}$ is directly related to the intrinsic
ionizing photon rate and thus, to the SFR, and the electron temperature (see
e.g., Eq. 10 & 11 in Murphy et al., 2011). For normal star-forming galaxies,
the non-thermal emission is dominated by CRs which were accelerated by
supernova and diffuse in the ISM. Thus, the non-thermal emission is directly
related to the supernova rate and $\alpha_{\mathrm{nth}}$(see Eq. 13 in Murphy
et al., 2011). The relative contribution of the non-thermal component in
normal star-forming galaxies increases as we go to lower frequencies (30 GHz
and below). We show the radio-SED, for a normal star-forming galaxy with
$f_{\mathrm{th}}=0.1$ (10%) at 1 GHz and $\alpha_{\mathrm{nth}}=-0.8$ in Fig.
2 as a dashed blue line. The dotted blue line shows the corresponding thermal
emission component. The combined effect of the steep non-thermal and flat
thermal component leads to an apparent spectral index between C- and S-bands
($\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$) equal to -0.64. In what follows, we
will refer to such a radio-SED and its apparent spectral index between C- and
S-bands as standard/canonical representing normal star-forming galaxies. We
will use this reference value of $\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$ to
compare with the spectral index of LzLCS sources.

For young galaxies (ages $<10$ Myr), it is possible that the fth is higher
than 0.1 due to a lack of enough supernovae and/or flat CR spectrum (¿-0.5)
and thus closer to that found in supernova remnants (SNR) (e.g., Hunt et al.,
2004, 2005). Moreover, in dense starbursts as discussed before, FFA has an
effect on both the thermal and non-thermal emission and leads to a turnover in
the radio spectra below a characteristic frequency (Condon, 1992). This is
termed the turnover frequency ($\nu_{\mathrm{t}}$) and is the frequency at
which the free-free optical depth ($\tau_{\nu}$) reaches unity. The FFA
$\tau_{\nu}$ is proportional to the emission measure (EM),
$\tau_{\nu}$$\propto\mathrm{n}_{\mathrm{e}}^{2}\mathrm{L}$ (pc cm-6), where
$\mathrm{n}_{\mathrm{e}}$ is the electron number density and L is the path
length along the line of sight. Thus $\nu_{\mathrm{t}}$ is also directly
related to EM and increases with increasing EM. Following Clemens et al.
(2010) and Galvin et al. (2018), we can then parameterize $\tau_{\nu}$ using
$\nu_{\mathrm{t}}$, such that, $\tau_{\nu}$ $=(\nu/\nu_{\mathrm{t}})^{-2.1}$.
We can thus modify the simple radio-SED from Eq. 1 under FFA in the simplest
case as:

$S_{\nu}=(1-e^{-\tau_{\nu}})\left[A_{1}+A_{2}\left(\frac{\nu}{\nu_{\circ}}\right)^{0.1+\alpha_{\mathrm{nth}}}\right]\left(\frac{\nu}{\nu_{\circ}}\right)^{2}.$
(2)

In Fig 2 we show the effect of FFA with $\nu_{\mathrm{t}}$ at 1 GHz as orange
line. In more complex scenarios there can be multiple FFA components leading
to even more complex features in the radio-SED (e.g., Clemens et al., 2010;
Galvin et al., 2018).

Loss of CRs particularly due to synchrotron and inverse-Compton (IC)
mechanisms can lead to curvatures in the spectrum towards high frequencies
(Klein et al., 2018). We show such a curved spectrum in Fig. 2 with the red
line. Finally, in the scenario of a single injection of CRs there can be a
sharp cutoff at higher frequencies in the CR energy spectrum leading to a
steepening at frequencies above a characteristic break frequency (cuttoff
model in Klein et al., 2018). We show such a spectrum with a cuttoff with the
green line in Fig. 2. We refer the reader to Klein et al. (2018) for a
detailed discussion on various modifications to the standard radio-SED. The
effect of steepening of the spectrum due to these mechanisms will show an
effect on the spectral index in the frequency range explored in this work
between (2-8 GHz), although we will need high-frequency (up to 30 GHz)
observations to distinguish between different scenarios.

Fig. 3 shows the observed RC SED for three LzLCS sources. In all three panels,
the blue dots show the observed flux densities, the gray line corresponds to a
simple radio-SED Eq. 1 normalized to the 3 GHz observed flux density with
$f_{\mathrm{th}}=0.1$ at 1 GHz and $\alpha_{\mathrm{nth}}=-0.8$ which is
typical of a star-forming galaxy. In the left panel is the radio-SED for
J144010+461937 which has a radio-SED close to a star-forming galaxy. In
Appendix C we can see that several sources in our sample show similar SEDs.

The middle panel shows the radio-SED for J095700+235709 where we see a
turnover below 3 GHz, and a steepening above 3 GHz. A mere radio-SED based
classification of such a source would term it to be a gigahertz peak spectrum
(GPS) or a high-frequency peakers (HFP) source (e.g., Gopal-Krishna et al.,
1983; Dallacasa et al., 2000; Stanghellini et al., 1998; O’Dea & Saikia, 2021;
Shao et al., 2022). The hosts of GPS and HFP sources are usually young AGNs.
However, we do not have any evidence of an AGN in these sources from the
optical emission lines (Flury et al., 2022a). We rather suggest that such a
radio-SED can arise in star-forming galaxies due to an effect of FFA below the
peak and steepening due to strong CR energy losses. The radio-SED of
J082652+182052 and J091113+183108 shown in Appendix C also shows evidence for
FFA based on the 3-$\sigma$ upper limits on the L-band flux densities.
Moreover, several sources shown in Appendix C are very steep at GHz
frequencies.

The right panel of Fig. 3 exhibits the radio-SED of J083440+480541 which is an
example of a broken SED above and below 3 GHz. J081409+211459 also shows a
similar feature (Appendix C). Such a SED can arise due to a mix of multiple
star-forming regions, one with an FFA component at high ($\sim$ 3 GHz), and
the other with a steep spectrum. Multiple ULIRGs in the Clemens et al. (2010)
sample show such a radio-SED. Interestingly, we also find that these two
sources are outliers when compared to their LyC properties which we discuss in
Sec. 4. Fig. C shows the radio-SED for our entire sample of C- and S-band
detections. Modelling the radio-SED to derive the relevant physical parameters
has several free parameters and requires a flux density measurement across a
large frequency range in order to remove the various degeneracies in parameter
estimation (see e.g., Galvin et al., 2018). Future observations of the radio-
SED at lower frequencies (¡ 1 GHz, e.g., with the Giant Meterewave Radio
Telescope and Low-Frequency Array) should help in constraining some of these
physical mechanisms.

###  3.2 Total spectral index distribution

Figure 4: Distribution of the total spectral index of LzLCS sources which
spans a wide range from flat to steep. The dashed line shows the value of
$\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$ expected in normal star-forming
galaxies with a $f_{\mathrm{th}}=0.1$ at 1 GHz and
$\alpha_{\mathrm{nth}}=-0.8$.

We calculate the total spectral index using the slope of the observed flux
densities at 6 and 3 GHz ($\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$) for our
sources. We propagate the 1-$\sigma$ flux uncertainties on the flux
measurements to estimate the uncertainties on the spectral index. For sources
with non-detections in only one of the S- and C-band, we estimate the
corresponding lower/upper limit on the spectral index using the $3\sigma$
upper limit on the flux density. In Table 1 we present the
$\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$ values for our sample. From our
sample of 24 C- and S-band detections, we could not derive reliable
$\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$ for three sources due to unreliable
flux density measurements. We use the rest of the sample of 21 sources with
reliable $\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$ for all further analysis
involving spectra indices. Overall in this sample 13 sources show steep
($\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$$<-0.64$) and 8 show flat radio
spectra ($\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$$\geq-0.64$) between 2-8 GHz.
For 7 of these sources we have L-band observations. Of them, 4 sources show a
turnover at frequencies below 3 GHz, which is evidence for a FFA component
between 1-3 GHz. And 2 sources show a broken radio-SED where the spectral
index between 1-3 GHz (L- and S-bands) is steeper than between 3-8 GHz (C- and
S-bands). The remaining one source shows a typical radio-SED of a normal star-
forming galaxy.

Fig. 4 shows the distribution of $\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$ in
the blue histogram. The mean $\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$ for our
sample is $-0.87$ with a scatter of 0.65. The total spectral index also varies
in a very broad range from $-2.18$ to 0.22. On the contrary, the mean total
spectral index of normal star-forming galaxies from the KINGFISHER galaxies
was found to be $-0.79$ with a scatter of 0.15 (Tabatabaei et al., 2017). Thus
LzLCS galaxies overall have a steeper mean radio spectra with a larger scatter
than normal star-forming galaxies.

Figure 5: LzLCS sources on the RC-SFR relation at 3GHz top panels and 6GHz in
the bottom panels. The left (right) panel shows the UV-based (H$\beta$-based)
SFRs for the LzLCS sources. In all panels, the blue points represent
detections in the corresponding band and the orange points represent the
3$\sigma$ upper limits for non-detections. The solid black line is plotted for
the standard RC-SFR relation with $\alpha_{\mathrm{nth}}{}=-0.8$. The dotted,
dashed and dotted dashed lines are for different values of
$\alpha_{\mathrm{nth}}$ from flat ($-0.3$) to very steep ($-2$) respectively.
The red solid line represents the RCth-SFR relation from Eq. 3. In all these
cases we have used a fixed value of T${}_{e}=10000$ K.

###  3.3 The RC-SFR relation for LzLCS sources

Figure 6: The ratio between the observed RC luminosity and that expected from
the RC-SFR relation from a standard radio-SED with
$\alpha_{\mathrm{nth}}=-0.8$, $f_{\mathrm{th}}=0.1$ at 1 GHz, and Te=104K, at
3 GHz and 6 GHz (top and bottom panels respectively). The left (right) panel
shows the ratio with the expected RC for the UV-based (H$\beta$-based) SFR.
The dashed line shows a ratio of 1 for reference.

The RC of normal star-forming galaxies at GHz frequencies can be used to
determine their SFR (Kennicutt & Evans, 2012; Murphy et al., 2011; Tabatabaei
et al., 2017). In this section, we compare the observed RC of LzLCS galaxies
to those predicted by such a calibration. The standard method used to
calibrate the RC-SFR relation relies on the tight radio (at 1.4 GHz)-Far-
infrared (FIR) correlation (de Jong et al., 1985; Helou et al., 1985).
However, since our observations are made at 3-6 GHz we need to scale to the
1.4 GHz luminosity assuming a spectral index. Since LzLCS sources show a large
variety in their radio-SED, such an assumption may not be valid across such a
large frequency range.

An alternate approach as discussed in Murphy et al. (2011) relates the thermal
RC (RCth) luminosity at a given frequency, $\nu$, to the SFR, as follows:

$\mathrm{SFR}^{\mathrm{th}}_{\nu}=4.6\times
10^{-28}\left(\frac{T_{e}}{10^{4}\leavevmode\nobreak\
\mathrm{K}}\right)^{-0.45}\left(\frac{\nu}{\mathrm{GHz}}\right)^{0.1}\left(\frac{L^{\mathrm{th}}_{\nu}}{\mathrm{ergs/s/Hz}}\right),$
(3)

where Te is the electron temperature. This is because both the SFR and RCth
emission are proportional to the ionizing photon rate in an optically thin
plasma. Moreover, the non-thermal RC (RCnth) luminosity at a given frequency,
$\nu$, can also be related to the SFR. This is because the SFR in a galaxy is
related to the SN rate and so is the synchrotron emission at GHz frequencies.
Therefore owing to a calibration between the SN rate and non-thermal
luminosity (Condon & Yin, 1990) we can derive a RCnth-SFR calibration (see Eq.
15 in Murphy et al., 2011). Here, it is assumed that the calibration between
SN rate and non-thermal relation is the same as that observed in our Galaxy.
These two relations can then be combined to relate the total RC luminosity and
the SFR as follows:

$\mathrm{SFR}_{\nu}=10^{-27}\left[2.18\left(\frac{T_{e}}{10^{4}\leavevmode\nobreak\
\mathrm{K}}\right)^{0.45}\left(\frac{\nu}{\mathrm{GHz}}\right)^{-0.1}\right.+\\\
\left.15.1\left(\frac{\nu}{\mathrm{GHz}}\right)^{\alpha_{\mathrm{nth}}}\right]^{-1}\left(\frac{L_{\nu}}{\mathrm{ergs/s/Hz}}\right).$
(4)

Using this relation is more convenient than the radio-FIR correlation to
compare galaxies with a non-standard $\alpha_{\mathrm{nth}}$ and thermal
fraction.

In Fig. 5, the solid black lines show the total RC-SFR calibration from Eq. 4
in all the panels. Here, we assumed a standard $\alpha_{\mathrm{nth}}=-0.8$
and T${}_{e}=10000$ K. We also show how this relation changes purely if the
$\alpha_{\mathrm{nth}}$ steepens to $-1.15$ and $-2$. Steep values of
$\alpha_{\mathrm{nth}}\sim-2.1$ are observed in nearby BCDs (Klein et al.,
1991), and hence we choose this range. Notice that the relation has the same
slope but it systematically moves down as the $\alpha_{\mathrm{nth}}$
steepens. This happens because, for a fixed frequency, a steeper non-thermal
spectrum has a lower contribution from the non-thermal emission. On the
contrary, a flatter $\alpha_{\mathrm{nth}}$ moves the relation upwards. In the
case where there is no non-thermal emission i.e., purely thermal radio
spectra, we will have the RCth-SFR relation from Eq. 3. The red solid lines in
both panels show this limit. In principle, galaxies can also be below this
line if the thermal emission is reduced due to FFA around 3 GHz. Such FFA at
GHz frequencies is observed in compact starbursts with high EM (e.g., Hunt et
al., 2004). We compare the RC luminosity with the SFR derived from two
different tracers, UV-based and H$\beta$-based. H-$\beta$ emission arises from
reprocessed ionizing radiation and is emitted by the same region as that of
the thermal radio emission. And the UV emission is simple starlight arising
from massive OB stars. The timescale of star-forlation traced by the H-$\beta$
line is of the order of 10 Myrs and is much shorter than the star-formation
timescale traced by the UV emission which is $\sim 100$ Myrs (Kennicutt &
Evans, 2012).

In Fig. 5, we show the RC luminosity at 3 GHz for LzLCS sources against the
extinction corrected UV-based SFR in the left panel and the H$\beta$-based SFR
in the right panel. For converting the observed S-band flux density of LzLCS
sources to the RC luminosity at 3 GHz for the sample with S-band detections we
used the observed total $\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$, and for the
non-detections we assumed a total
$\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$$=-0.64$, as that of normal galaxies.
As such there is a relatively weak dependence of the spectral index on the
luminosities as the K-corrections are small since our sources are at a
relatively low redshift of $\sim 0.3$. The blue points in both panels are for
the C- and S-band detection sample, and the orange points are the 3$\sigma$
upper limits for the sample with S-band non-detections. We see that although
LzLCS sources show a correlation between their RC luminosity and SFR, they
show a lot of scatter from the standard RC-SFR relation, with several sources
much above and below this relation. It is important to note that several
points are above the RCth-SFR relation, thus suggesting that LzLCS sources
have non-thermal emission which suggests past SN activity.

Figure 7: Dependence of $f_{\mathrm{esc}}^{\mathrm{LyC}}$ on the total
spectral index measured across C- (6 GHz) and S- (3 GHz) bands. The left panel
shows the dependence on $f_{\mathrm{esc}}^{\mathrm{LyC}}$ measured using the
UV-fit method. And the right panel shows the dependence on
$f_{\mathrm{esc}}^{\mathrm{LyC}}$ measured using the H$\beta$ method Flury et
al. (2022a). The blue points in both panels show LyC detections and the orange
points show LyC non-detections (1$\sigma$ upper limits). We also show two
known nearby leakers from the literature, Haro 11 (brown star) and Tol
1247-232 (black star). In both panels the horizontal lines demarcate the
regions for strong leakers ($f_{\mathrm{esc}}^{\mathrm{LyC}}$ ¿ 0.1), weak
leakers ( 0.01 ¡ $f_{\mathrm{esc}}^{\mathrm{LyC}}$ ¡ 0.1), and non-leakers
($f_{\mathrm{esc}}^{\mathrm{LyC}}$ ¡ 0.01). The vertical line shows the region
above and below a spectral index of a standard radio-SED
($\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$$=-0.64$) demarcating the region for
flat and steep spectrum sources. See text for details.

We next investigate if the offset between the observed and expected RC
luminosity can be explained by variations in the observed
$\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$ of these sources. In Fig. 6, we show
the ratio between the observed RC luminosity and that expected from the RC-SFR
relation based on Eq. 4 (termed as the “SFR offset” here on) with respect to
the total $\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$ for only the C- and S-band
detection sample. We study the RC-offset for both the S- and C-bands and for
SFR measured using UV and H$\beta$ emission. We notice that the H$\beta$-based
SFR offset systematically increases for steep spectrum sources (top right
panel) for the S-band, whereas the flat spectrum sources are closer to that
expected. The SFR offsets with UV-based SFRs at C-band are higher for flat
spectrum sources (bottom left panel). Such a deviation can lead to values
above the expected RC-SFR relation in Fig. 5. The SFR offsets with UV-based
SFRs for S-band (top left panel) and H$\beta$-based SFRs for C-band (bottom
right panel) do not show any such offsets although they have a large scatter.
We currently do not completely understand the origin of such systematic and
large offsets. Overall this suggests that accurate SFR estimation using
different tracers is difficult for LzLCS sources and particularly so with
radio-based SFR tracers.

Finally, an important reason for the scatter in the RC-SFR relation can be
that for galaxies underdoing starbursts like the LzLCS sources, the SFR traced
by different methods are in general not consistent due to the
stochastic/bursty nature of star formation (e.g., Izotov et al., 2016a; Atek
et al., 2022). This inconsistency is also evident between the UV-based and
H$\beta$-based SFRs. Finally, for the leakers (weak and strong) the
H$\beta$-based SFR can underestimate the true SFR, since not all the ionizing
photons are absorbed by the star-forming regions. Most of the leakers in our
sample are weak ($<10$%) and the two strong leakers have
$f_{\mathrm{esc}}^{\mathrm{LyC}}$ below $30$%. Thus this effect is marginal
compared to the large SFR offsets we observe. We also see that the RC is
significantly below the RC-SFR relation in several sources. This can happen
due to a combination of a steep $\alpha_{\mathrm{nth}}$ or a relatively low
non-thermal fraction. In a few sources, their RC luminosity is fairly close to
that expected from the RCth-SFR relation, and are thus possibly thermally
dominated systems. Such a deficit has also been observed in a sample of green-
peas (Chakraborti et al., 2012), blueberries (Sebastian & Bait, 2019) and in
Hii galaxies (Rosa-González et al., 2007). The effect of FFA on both thermal
and non-thermal emissions can also be a further cause of such a deficit. A few
sources are above the black solid line, where it is possible that their
$\alpha_{\mathrm{nth}}$ is flatter than the canonical value. It is also
possible that the calibration between the SN rate-SFR observed in our Galaxy
is not valid for LzLCS sources.

Figure 8: Dependence of spectral index
$\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$ on $\beta_{1550}$, O32 and
$\Sigma_{\mathrm{SFR}}$ for the LzLCS sample. The vertical line demarcates the
flat ($\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$$>-0.64$) and steep spectrum
($\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$$<-0.64$) sources.

###  3.4 Relations between LyC escape and RC emission

In this section, we for the first time empirically study the dependence of the
Lyman continuum escape fraction, $f_{\mathrm{esc}}^{\mathrm{LyC}}$, on the
total spectral index and thermal fraction. Figure 7 shows the dependence of
$f_{\mathrm{esc}}^{\mathrm{LyC}}$ on the total spectral index measured in the
two bands, C- and S. For this and all further analysis we use only the sources
with reliable $\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$ for our analysis. The
sources with LyC detections are shown in blue points and the non-detections
are shown as upper limits (1-$\sigma$) in orange points.

The left and right panels shows the radio spectral index dependence on
$f_{\mathrm{esc}}^{\mathrm{LyC}}$ measured using two different approaches,
named the “UV-fit” and “H$\beta$” methods (see Flury et al., 2022a, for
details). In both methods, the escape fraction is calculated as the ratio
between the _observed_ (escaping) and the _intrinsic_ (produced) ionizing
fluxes, both measured at a narrow wavelength interval just below the Lyman
edge (912Å). Both former approaches differ according to the way the latter
intrinsic ionizing flux is estimated. In the UV-fit method, the _intrinsic_
LyC flux was predicted via detailed SED-fitting to non-ionizing part of the
COS/G140L spectra of LzLCS galaxies (see Saldana-Lopez et al., 2022). In the
H$\beta$ method, the _intrinsic_ LyC flux density was estimated from the
measured equivalent-width of the H$\beta$ line from SDSS spectra, as a tracer
of the age of the most massive stellar populations (see also Izotov et al.,
2016c). In instantaneous-burst models (Leitherer et al., 1999, used
consistently for both former approaches), the age of the burst is inherently
related to the production of LyC photons.

The horizontal dashed lines show the region for strong leakers
($f_{\mathrm{esc}}^{\mathrm{LyC}}$ ¿ 0.1), weak leakers ( 0.01 ¡
$f_{\mathrm{esc}}^{\mathrm{LyC}}$ ¡ 0.1), and non-leakers
($f_{\mathrm{esc}}^{\mathrm{LyC}}$ ¡ 0.01). The vertical dashed line shows the
value of $\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$ for a standard radio-SED
from Eq. 1 with a $\alpha_{\mathrm{nth}}=-0.8$ and $f_{\mathrm{th}}=0.1$ at 1
GHz. This line splits the figure into regions with flat spectrum
($\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$ $>-0.64$) and steep spectrum sources
($\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$ $<-0.64$). We also show two known
nearby weak leakers, Haro 11 and Tol 1247-232 on this plot. We used the
literature value of $f_{\mathrm{esc}}^{\mathrm{LyC}}$ from Leitet et al.
(2011) and Leitherer et al. (2016) for Haro 11 and Tol 1247-232 respectively.
The radio spectral index for Haro 11 was measured between 3.0-8.46 GHz. We
used Schmitt et al. (2006) for the 8.46 GHz flux density and the Very Large
Array Sky Survey (VLASS) catalogue (Gordon et al., 2021) for the 3.0 GHz flux
density. The radio spectral index for Tol 1247-232 was measured between 5-8
GHz using VLA observations from Rosa-González et al. (2007). The radio
spectral index is measured in a very similar range frequency range as our C-
and S-bands allowing us to use them for comparison purposes.

It is interesting to find a positive correlation between
$f_{\mathrm{esc}}^{\mathrm{LyC}}$, particularly measured using the UV-fit and
$\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$. The correlation appears to be
marginally weak for the $f_{\mathrm{esc}}^{\mathrm{LyC}}$ measured using
H$\beta$, however, that seems to be largely driven by the difference in the
upper limits on the LyC non-detections between the two methods. This indicates
that strong leakers show overall flatter radio spectra, and non-leakers have
steeper $\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$, while weak leakers typically
have $\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$ close to the canonical value
($\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$ $=-0.64$). Interestingly, Haro 11
and Tol 1247-232 are also within our
$\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$-$f_{\mathrm{esc}}^{\mathrm{LyC}}$
relation. We also find two outliers (J155945+403325 and J083440+480541) from
the $\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$-$f_{\mathrm{esc}}^{\mathrm{LyC}}$
relation (in both the plots) and J081409+211459 which is clearly an outlier in
the left-panel of Fig. 7 and possibly in the right panel. These sources show a
flat spectrum but much lower values of $f_{\mathrm{esc}}^{\mathrm{LyC}}$ than
other LzLCS sources. Interestingly, J083440+480541 and J081409+211459 have
L-band observations and show a steeper spectrum at low frequencies. However,
in other sources with L-band observations we see either a lower L-band flux
density or that consistent with a standard radio-SED. Finally in Table 3 we
quantify these correlations using the Kendal $\tau$ correlation coeffiecient.
Here we take into account the upper limits on the
$f_{\mathrm{esc}}^{\mathrm{LyC}}$ values following Akritas & Siebert (1996)
which is the procedure following in Flury et al. (2022b). We find that
$\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$ correlates with both
$f_{\mathrm{esc}}^{\mathrm{LyC}}$ UV-fit and H$\beta$ method (strong $\tau$
values) particularly, after removing the two outliers. However the
significance of this correlation is less compared to other diagnostics
discussed in Flury et al. (2022b). It is important to note that our current
sample is quite small compared to the full sample studied in Flury et al.
(2022b) and also lacks sources at high LyC escape. In the future, we aim to
increase the sample size with radio observations of LyC leakers, and
particularly with L-band data. This should give us a better picture of the
dependance of $f_{\mathrm{esc}}^{\mathrm{LyC}}$ and
$\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$ and on the properties of outliers.

###  3.5 Relation between $\beta_{1550}$, O32, and $\Sigma_{\mathrm{SFR}}$ on
the RC emission

The escape of LyC photons from galaxies is known to be correlated with several
other physical parameters from the literature (e.g., Izotov et al., 2021b;
Flury et al., 2022b; Saldana-Lopez et al., 2022; Chisholm et al., 2022;
Marques-Chaves et al., 2022b; Xu et al., 2023), particularly the UV slope
$\beta_{1550}$, the O32 emission line ratio, and SFR surface density
$\Sigma_{\mathrm{SFR}}$. In particular, galaxies with high
$f_{\mathrm{esc}}^{\mathrm{LyC}}$ are expected to have steeper values of
$\beta_{1550}$, higher O32, and a higher $\Sigma_{\mathrm{SFR}}$ and vice
versa. Thus the correlation between $\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$
and $f_{\mathrm{esc}}^{\mathrm{LyC}}$ observed here leads us to explore if the
$\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$ of LzLCS sources also depends on
these parameters with similar trends. Moreover, the LzLCS sample is also
selected based on a range of $\beta$, O32, and $\Sigma_{\mathrm{SFR}}$. In
Fig. 8, we show the dependence of $\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$ on
$\beta_{1550}$, O32, and $\Sigma_{\mathrm{SFR}}$. We see that
$\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$ and $\beta_{1550}$ are overall anti-
correlated such that the $\beta_{1550}$ slope steepens as
$\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$ flattens. The presence of flatter
$\beta_{1550}$ slopes (i.e. redder UV spectra) in sources with steep
$\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$ can be associated with the declining
phase of the starburst (see Sec. 4.3 for more discussion) and also to higher
dust attenuation (e.g., Saldana-Lopez et al., 2022; Chisholm et al., 2022). We
also notice that $\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$ overall flattens
with increasing O32 and $\Sigma_{\mathrm{SFR}}$ which is both in line with
that expected with their dependence on $f_{\mathrm{esc}}^{\mathrm{LyC}}$.
However, it is difficult to associate a direct causal relation between
$\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$ and these parameters as
$f_{\mathrm{esc}}^{\mathrm{LyC}}$ depends on multiple physical parameters
(e.g., Flury et al., 2022b). We report the Kendall $\tau$ values between these
quantities in Table 3. We find a strong correlation between these quantities
but with a low significance (low $p-$values). We discuss the possible physical
reasons driving a correlation between $\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$
and $f_{\mathrm{esc}}^{\mathrm{LyC}}$ in the next section.

Table 3: Kendall $\tau$ correlation coefficients and the $p-$values between
the radio spectral index ($\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$) versus
$f_{\mathrm{esc}}^{\mathrm{LyC}}$ and different physical properties of LzLCS
sample. Here we take into account the upper limits on
$f_{\mathrm{esc}}^{\mathrm{LyC}}$ (following Akritas & Siebert (1996)) with
the same approach as described in Flury et al. (2022b).

| $\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$
---|---
| $\tau$ | $p_{val}$
$f_{\mathrm{esc}}^{\mathrm{LyC}}$(UV-fit)∗ | $+0.386(0.277)$ |  $1.445\times 10^{-2}$ ($6.450\times 10^{-2}$)
$f_{\mathrm{esc}}^{\mathrm{LyC}}$(H$\beta$)∗ | $+0.371(0.253)$ |  $1.85\times 10^{-2}$ ($9.098\times 10^{-2}$)
$\beta_{1550}$ | $-0.400$ | $1.120\times 10^{-2}$
$\log_{10}$O32 | $+0.429$ | $6.573\times 10^{-3}$
$\log_{10}$$\Sigma_{\mathrm{SFR}}$(H$\beta$) | $+0.286$ | $7.000\times 10^{-2}$
444∗ Here the $\tau$ values are estimated after removing the two outliers and
the values in the bracket are including the outliers. The typical uncertainty
in $\tau$ is $\pm$0.11.

##  4 Discussion

###  4.1 Estimated thermal fraction of LzLCS sources

Figure 9: Distribution of the estimated thermal fraction for LzLCS sources.
The blue (orange) histograms show the thermal fraction at 3 GHz (6 GHz). The
solid (dashed) line shows the expected thermal fraction at 3 GHz (6 GHz) for
normal star-forming galaxies with a $f_{\mathrm{th}}=0.1$ at 1 GHz and
$\alpha_{\mathrm{nth}}=-0.8$.

Figure 10: Dependence of the estimated thermal fraction at 3 GHz on
$f_{\mathrm{esc}}^{\mathrm{LyC}}$ measured using the UV-fit. The horizontal
dashed lines demarcate the region for strong, weak and non-leakers. The
vertical dashed line shows the expected thermal fraction at 3 GHz for normal
star-forming galaxies with a $f_{\mathrm{th}}=0.1$ at 1 GHz and
$\alpha_{\mathrm{nth}}=-0.8$.

We estimated the thermal flux density at two different observed frequencies, 3
GHz and 6 GHz, using Eq. 3. We use the SFR (derived from the dust-corrected
H$\beta$ line intensity) and electron temperature derived using [OIII]
$\lambda\lambda 4363;4959,5007$ (see Flury et al., 2022a, for details). We
then subtract the thermal contribution at the two frequencies and estimate the
corresponding non-thermal flux density. The mean value of the thermal fraction
at 3 GHz (6 GHz) is 0.21 (0.36). The mean thermal fraction at 6 GHz for nearby
star-forming galaxies from the KINGFISHER sample was found to be $0.23\pm
0.13$ Tabatabaei et al. (2017) which is slightly higher although within the
uncertainties. Moreover, the range of the thermal fractions of our sample is
similar to that of the KINGFISHER sample. Fig. 9 shows the distribution of the
thermal fraction at 3 GHz in blue and 6 GHz in the orange histogram. The fact
that the radio emission from these galaxies is not dominated by thermal
emission provides evidence for a non-zero contribution from synchrotron
emission in these galaxies. Interestingly, this provides the first direct
evidence of SN activity in these galaxies.

Next, we investigate if SN activity could play a role in facilitating LyC
escape as expected from various simulations (e.g., Trebitsch et al., 2017; Ma
et al., 2020). In Fig. 10 we study the dependence of
$f_{\mathrm{esc}}^{\mathrm{LyC}}$ on the thermal fraction which informs us
about increased SN activity or the lack of it. We do not observe any
correlation, rather most of our sources including the strongest leakers have
thermal fractions close to that expected for normal star-forming galaxies
(vertical dashed line). Most of the non-leakers instead are non-thermal
dominated thus suggesting older episodes of star formation. A similar lack of
dependence is observed between the thermal fraction derived at 6 GHz and
$f_{\mathrm{esc}}^{\mathrm{LyC}}$.

A few sources in our sample are very compact starbursts where it is common to
have a significant FFA component. Therefore, it is possible that the true
thermal fractions can be even higher than our estimates. A complete radio-SED
fitting, particularly with even higher frequency data, is required to
accurately constrain the thermal fractions.

###  4.2 Why do strong leakers have a flat radio spectrum?

Figure 11: Effect of increasing fth at a reference frequency of 3 GHz (top
panel) and 6 GHz (bottom panel) on the observed spectral index
$\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$. Here we consider the simplest radio-
SED composed of a thermal and non-thermal component as shown in Eq. 1. The
different coloured lines show the effect of increasing the
$\alpha_{\mathrm{nth}}$ from the canonical value of $-0.8$ to up to $-1.5$. In
both the panels the dashed horizontal lines show the value of
$\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$$=-0.64$ for $f_{\mathrm{th}}=0.1$,
and $\alpha_{\mathrm{nth}}=-0.8$ from Eq. 1. And the dashed vertical line
shows the expected value of the thermal fraction at 3 and 6 GHz for such a
radio spectrum. We also show our LzLCS observations on this plot in blue
points.

As seen in Fig. 7, the observed spectral index across the C- and S-bands,
$\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$, flattens with increasing
$f_{\mathrm{esc}}^{\mathrm{LyC}}$. This hints towards a flattening of
$\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$ due to young stellar ages of high
$f_{\mathrm{esc}}^{\mathrm{LyC}}$ galaxies ($f_{\mathrm{esc}}^{\mathrm{LyC}}$
$>0.1$). Young stellar systems (ages $\leq 5$Myrs) will be undergoing some of
the first events of SN, which would in turn result in a higher thermal
fraction. This will lead to a gradual flattening of
$\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$ with decreasing stellar age. In Fig.
11, we show the dependence of the estimated thermal fraction at 3 GHz (top
panel) and 6 GHz (bottom panel) on the
$\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$ for our LzLCS sample. We
quantitatively show the effect of increasing the fth, defined at a reference
frequency of 3 GHz (top panel) and 6 GHz (bottom panel), on the observed
$\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$. This is calculated using the simple
model for the radio-SED consisting of a thermal and a non-thermal component as
described in Eq. 1. The blue line shows the variation for a canonical value of
$\alpha_{\mathrm{nth}}=-0.8$. We also show the effect of steepening
$\alpha_{\mathrm{nth}}$ to up to $-1.5$ in different coloured lines. The
dashed horizontal line shows the value of
$\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$$=-0.64$ which is expected for a
standard $f_{\mathrm{th}}=0.1$ at 1 GHz, and $\alpha_{\mathrm{nth}}=-0.8$. The
dashed vertical line shows the value of the expected thermal fraction at 3 and
6 GHz for such a model.

As expected in all the models with both flat and steep values of
$\alpha_{\mathrm{nth}}$, $\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$ flattens as
we increase fth. Almost half of our sample lies within the region bracketed by
these models. Thus their variation in the spectral indices in principle can be
associated with their different values of fth and $\alpha_{\mathrm{nth}}$.
However, very flat spectrum sources ($\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$
$>-0.5$) would need other mechanisms to further flatten the spectrum e.g., FFA
as discussed below. Similarly very steep spectrum sources
($\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$ $<-1.5$) would either need unusually
steep $\alpha_{\mathrm{nth}}$, or other mechanisms to steepen
$\alpha_{\mathrm{nth}}$, e.g., in a break model or CR escape which we discuss
in Sec. 4.3.

Figure 12: Effect of increasing the turnover frequency ($\nu_{\mathrm{t}}$) on
$\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$. The solid lines are for a canonical
$\alpha_{\mathrm{nth}}{}$=-0.8, with the blue line corresponding to a
$f_{\mathrm{th}}{}=0.1$. The other lines shows the effect of increasing the
fth. The dashed lines are for the same range of fth but with a more steep
$\alpha_{\mathrm{nth}}=-2$. The dashed horizontal line shows a value of
$\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$$=0.$ This shows that we need FFA with
a $\nu_{\mathrm{t}}$ of $\gtrsim 1.5$ GHz to flatten
$\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$ $>0$.

FFA also has a strong effect on flattening the radio-SED. In normal galaxies,
the effect of FFA is usually observed at low frequencies below $\sim 30$MHz
(Chyży et al., 2018). For dense starbursts, FFA can be important at relatively
higher frequencies around $\sim 700$MHz for starburst galaxies like in M82
(see e.g., Chyży et al., 2018) and close to GHz for ULIRGS (e.g., Clemens et
al., 2010; Galvin et al., 2018). However, for low metallicity BCDs, for
example in SBS 0335-052, which host a dense super star cluster, Hunt et al.
(2004) observed a turnover frequency between 2-3 GHz, corresponding to an EM
of $\sim 8\times 10^{7}$pc cm-6. In Fig. 12, we show the effect of increasing
$\nu_{\mathrm{t}}$, from 10 MHz to $\sim$5 GHz, on the observed
$\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$. We also show the effect of
increasing the fth from 0.1 to 1.0 on $\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$
shown in solid lines. In dashed lines, we show the same set of curves with the
effect of steepening the $\alpha_{\mathrm{nth}}$ to $-2.0$. Notice that in
order to flatten $\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$ $>0.0$, in all the
models we need a $\nu_{\mathrm{t}}$ $\gtrsim 1.5$GHz, which corresponds to an
EM $\sim 1.2\times 10^{7}$ pc cm-6 assuming a $z\sim 0.3$ to convert the
observed $\nu_{\mathrm{t}}$ to the rest-frame and the Te of $10^{4}$ K. This
is also in line with the correlation seen between
$f_{\mathrm{esc}}^{\mathrm{LyC}}$ and $\Sigma_{\mathrm{SFR}}$ (e.g., Izotov et
al., 2016b; Naidu et al., 2020; Flury et al., 2022b). Hence, increasing
$\Sigma_{\mathrm{SFR}}$ (and thus the EM) can indirectly lead to an increase
in $\nu_{\mathrm{t}}$ and thus flatten
$\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$. This can become particularly
important for the high $f_{\mathrm{esc}}^{\mathrm{LyC}}$ which has the highest
$\Sigma_{\mathrm{SFR}}$. Finally, for normal star-forming galaxies in the
KINGFISHER sample, it is observed that $\alpha_{\mathrm{nth}}$ flattens with
increasing $\Sigma_{\mathrm{SFR}}$ (Tabatabaei et al., 2017). Such an effect
is also observed in the arms and inner regions within a galaxy where the gas
density is higher and the $\alpha_{\mathrm{nth}}$ can flatten to even higher
than -0.5 (Basu et al., 2015). More importantly in the case of IZw 18, a
nearby low-metallicity BCD, Hunt et al. (2005) finds a rather flat
$\alpha_{\mathrm{nth}}\approx-0.5$. This is associated with the CR energy
index being close to the injection spectrum during SN acceleration (Tabatabaei
et al., 2017; Basu et al., 2015). Our LzLCS sources are several orders of
magnitude denser than KINGFISHER sample leading to the possibility that they
too have relatively flat $\alpha_{\mathrm{nth}}$ which can overall flatten
$\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$.

We conclude that since high $f_{\mathrm{esc}}^{\mathrm{LyC}}$ galaxies have a
relatively young age of the starburst and relatively high
$\Sigma_{\mathrm{SFR}}$ they can have a high thermal fraction, EM, and
possibly a flatter $\alpha_{\mathrm{nth}}$. These effects can altogether
flatten the $\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$ which can explain the
observed trend for strong leakers in Fig. 7. Finally, a flat spectrum source
is not always a strong leaker source. We have observed a few outliers in the
$\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$-$f_{\mathrm{esc}}^{\mathrm{LyC}}$
relation. This indicates flat spectrum is possibly a necessary but not a
sufficient condition for identifying strong leakers.

###  4.3 Why do non-leakers have a steep radio spectrum?

Figure 13: Ratio between the SFR derived using UV and H$\beta$ luminosity as a
function of the EW(H$\beta$). Notice that typically galaxies with lower EW
show a higher UV-based SFR compared to H$\beta$ based SFR. We also overplot
the stellar population synthesis models from Schaerer (2003) for a single
burst model in solid lines, and continuous star-formation in dashed lines. The
different colours blue, orange, green, and red represent the different
metallicities corresponding to $Z=0.02$ (solar), 0.008, 0.004, and 0.001
respectively.

Figure 14: Ratio between the SFR derived using UV and H$\beta$ luminosity vs.
the $\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$ for our LzLCS sample. The
horizontal line represents a ratio of 1. And the vertical line shows the
$\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$ for a canonical radio spectrum.
Notice that typically galaxies with a higher UV-based SFR compared to H$\beta$
based SFR (and also lower EW as seen in Fig. 13) show a steeper spectra.

It is quite remarkable that the correlation between
$\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$ and $f_{\mathrm{esc}}^{\mathrm{LyC}}$
extends all the way towards non-leakers, and in particular, non-leakers have
very steep $\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$ ($<-1$). Several processes
related to either the complete loss of CRs or energy-dependent losses are
usually invoked to explain the steepening of
$\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$, for example, a break, cutoff, or
curvature at higher frequency in the radio spectrum (e.g., see the models
listed in Klein et al., 2018). A break in the spectrum is produced in an open-
box model where the CRs can escape the halo of the galaxy (Lisenfeld et al.,
2004). This break occurs due to a high convective speed due to which the CRs
can leave the halo (e.g., Lisenfeld et al., 2004; Klein et al., 2018). The
break steepens $\alpha_{\mathrm{nth}}$ by 0.5 above a characteristic frequency
termed as the break frequency (Klein et al., 2018). The break frequency is
related to the convection speed and size of the halo. Thus, an original
$\alpha_{\mathrm{nth}}\sim-0.8$ can easily steepen to $\sim-1.3$ above the
break frequency. A cutoff in a spectrum occurs when there is an abrupt change
in the CR energy spectrum arising due to a relatively recent starburst a few
Myrs ago. This happens in a so-called “single-injection scenario” where most
of the CRs were produced in a recent starburst. The high-energy CRs will lose
their energy faster than the low-energy ones, and it will produce a strong
break above some CR energy scale (Schlickeiser, 1984; Klein et al., 2018).
This CR break energy is related to the break frequency and depends inversely
on the time of CR injection (see Eq. 4 in Lisenfeld et al., 2004).

Breaks in the radio spectrum were observed in NGC 1569, a nearby dwarf
starburst, at around 8 GHz (Israel & de Bruyn, 1988), and several other nearby
dwarf galaxies at GHz frequencies ranging from 1-12 GHz (Deeg et al., 1993;
Klein et al., 2018). Israel & de Bruyn (1988) applied the single-injection CR
acceleration model to NGC 1569 and estimated the injection time to $\sim$5
Myrs ago. Lisenfeld et al. (2004) has interpreted the same break in NGC 1569
due to a single-injection CR model (also termed as the cutoff model) or due to
CR outflows, however, they favor the latter scenario. Deeg et al. (1993)
interpret several of their convex radio spectra in BCDs due to a break from a
single-injection CR model.

If non-leakers in the LzLCS sample are “the low-phase of a bursty SFR”, i.e.
have recently stopped star-formation or have a declining SFR with time, we
expect to see to first order differences in their SFRs derived using UV and
H$\beta$ luminosities, since the two trace SFR over long ($\sim$100 Myrs) and
short timescales ($\sim$10 Myrs) respectively (e.g., Kennicutt & Evans, 2012).
In Fig. 13 we compare the ratio
$\mathrm{SFR}{{}_{\mathrm{UV}}}/\mathrm{SFR}{{}_{\mathrm{H}\beta}}$ with the
EW(H$\beta$) for our sample. We notice that sources with the lowest
EW(H$\beta$) also have the highest ratio of
$\mathrm{SFR}{{}_{\mathrm{UV}}}/\mathrm{SFR}{{}_{\mathrm{H}\beta}}$, which is
naturally explained by relative age differences, with younger objects having
higher EW(H$\beta$). To demonstrate this, we overplot predictions from the
synthesis models of Schaerer (2003) for metallicities bracketing those of the
LzLCS and both for instantaneous burst models and constant SFR. Qualitatively
the observed trend of increasing
$\mathrm{SFR}{{}_{\mathrm{UV}}}/\mathrm{SFR}{{}_{\mathrm{H}\beta}}$ with
decreasing EW(H$\beta$), corresponding to a temporal evolution, is reproduced,
and models of instantaneous burst (or other declining SF histories) also
naturally lead to
$\mathrm{SFR}{{}_{\mathrm{UV}}}/\mathrm{SFR}{{}_{\mathrm{H}\beta}}$$>1$.

In Fig. 14, we compare this SFR ratio with the observed
$\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$ for our LzLCS sample. Notice that the
sources with the steepest spectra (thus non-leakers), show the highest ratio
of $\mathrm{SFR}{{}_{\mathrm{UV}}}/\mathrm{SFR}{{}_{\mathrm{H}\beta}}$. In
such a cutoff model we can also estimate the CR injection timescale Lisenfeld
et al. (using Eq. 4 from 2004) for a break frequency close to 6 GHz and with
magnetic fields ranging from 10-20 $\mu$G. We find that in such a model for
the entire range of magnetic fields, a break can occur below 6 GHz within 10
Myrs. This timescale is consistent with the timescale on which we expect to
see differences in the UV-based and H$\beta$-based SFR, and further supports
our idea that the steepening of our LzLCS sources is because of their
declining SFR with time.

Alternatively, it is possible that the steepening is caused due to CR escape
from advection or diffusion. In the intriguing case of the starburst dwarf
galaxy, IC 10, there is evidence of SN feedback in the form of a non-thermal
superbubble (Yang & Skillman, 1993). The spectral index is steep (close to
-1.0) and is found to be consistent with CR escape due to advection or
diffusion (Heesen et al., 2011, 2018). Strong outflows are also observed in
compact starbursts (e.g., Weiner et al., 2009; Heckman et al., 2011; Chisholm
et al., 2017). Given the typically small sizes of these galaxies, the
advection/diffusion timescales are expected to less than 1 Myr (Sebastian &
Bait, 2019). However, it is not directly very clear why the
advection/diffusion driven CR escape should be more prominent in non-leakers
as opposed to leakers. Farcy et al. (2022) showed that the presence of CRs in
simulated galaxies tend to smooth out the ISM thus filling it with denser gas
and hence reducing $f_{\mathrm{esc}}^{\mathrm{LyC}}$. In the future, the
inclusion of high and low-frequency radio data should better constrain the
radio-SED and help us in distinguishing between the break or cutoff model for
the steepening of $\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$.

###  4.4 Overall picture relating LyC escape and RC variations

In summary, since the radio spectral index is a proxy for the age of the
starburst (Cannon & Skillman, 2004; Hirashita & Hunt, 2006), we possibly
observe a relation between $\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$ and
$f_{\mathrm{esc}}^{\mathrm{LyC}}$. Using a timescale argument we propose the
following evolutionary picture. After the onset of a starburst (¡ 5 Myr)
typically in dense ISM conditions, the radio spectrum will be flat
($\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$ $>-0.65$) due to i) thermally
dominated radio emission as most of the young stars haven’t yet undergone
supernova ii) the less dominant non-thermal emission which would be arising
when some of the first supernovae start to explode will accelerate fresh CRs
that have a flat non-thermal spectrum (¿ -0.65) and iii) the presence of dense
ISM conditions can lead to a flat radio spectrum due to the strong effect of
FFA at GHz frequencies. These physical conditions which lead to a flat
$\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$ are also when the
$f_{\mathrm{esc}}^{\mathrm{LyC}}$ is highest. At later times (5-10 Myrs) when
the starburst is declining along with the $f_{\mathrm{esc}}^{\mathrm{LyC}}$,
high-energy CRs also lose their energies (in $\sim$10 Myrs) which results in
the steepening of $\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$ ($<-0.65$).
Consequently, the galaxies will quickly move down along the
$\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$-$f_{\mathrm{esc}}^{\mathrm{LyC}}$
relation in Fig. 7 in around 10 Myrs. If there are any further bursts of star
formation in these galaxies they can move along this sequence. Alternatively,
the steepening can also be caused due to CR escape. For galaxies showing
continuous star formation over 100 Myrs timescale, the CRs would have diffused
in the galaxy and have a non-thermal spectral index close to the canonical
value of -0.8 (Condon, 1992). Such galaxies would then have a total
$\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$ close to $-0.65$ and
$f_{\mathrm{esc}}^{\mathrm{LyC}}$$<1$%.

Finally, the fact that some properties of the radio continuum correlate with
the LyC escape fraction could indicate that the escape of ionizing photons is
not highly directional, i.e. happens in a relatively isotropic manner. This is
due to the fact that the observational determination of LyC escape
($f_{\mathrm{esc}}^{\mathrm{LyC}}$) measures a directional quantity, namely
the escape of ionizing photons along the line-of-sight towards the observer,
whereas RC emission is a priori a volumetric probe and orientation-
independent. Therefore the finding of a correlation between
$f_{\mathrm{esc}}^{\mathrm{LyC}}$ and RC continuum properties would indicate
that $f_{\mathrm{esc}}^{\mathrm{LyC}}$ is also orientation-independent, to
first order. This is consistent with previous results inferred from
correlations between global galaxy properties and
$f_{\mathrm{esc}}^{\mathrm{LyC}}$ (Wang et al., 2019). Now in reality the
relation between $\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$ and
$f_{\mathrm{esc}}^{\mathrm{LyC}}$, e.g., shows a considerable scatter, which
might partly be driven by some orientation-dependence. Better statistics, and
a more fundamental understanding of the origin of these correlations will
therefore be needed to properly address the question of non-isotropic LyC
escape. Other volumetric measures at other wavelengths also exist. For
example, the nebular emission lines in the optical are thought to be
essentially optically thin, and $f_{\mathrm{esc}}^{\mathrm{LyC}}$ is found to
correlate with O32, which could again indicate relatively isotropic LyC
emission. However, the scatter in these relations is also large (see Izotov et
al., 2021b; Flury et al., 2022b), and there is no unique and direct causal
connection between LyC escape and O32, which a priori depends on several
parameters, such as ionization parameter, metallicity, the age of stellar
population(s), etc., and $f_{\mathrm{esc}}^{\mathrm{LyC}}$ (Nakajima & Ouchi,
2013; Ramambason et al., 2020). Possible orientation and geometrical effects
of LyC escape are so far difficult to establish observationally from the
available low-$z$ galaxy samples.

##  5 Conclusions and future work

To study for the first time the radio continuum (RC) properties of galaxies
with LyC measurements, we have obtained VLA C- and S-band observations for a
sample of 53 galaxies. We have also obtained L-band observations for 17 of
these sources. These galaxies are selected from the recent Low-$z$ Lyman
Continuum Survey (LzLCS; Flury et al., 2022a, b) which contains a large sample
of low-$z$ ($z\sim 0.3-0.4$) star-forming galaxies with LyC measurements. We
have detected RC at both C- and S-bands for about half of our sample where we
then can measure the total spectral index
($\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$) and thermal fraction.

Our study highlights an interesting connection between the radio spectral
index and the escape of LyC photons. Overall we associate this with the
rapidly changing SFH (on a few Myr timescale) due to the onset of starburst
and its decline in LzLCS galaxies. These changes in SFHs are traced by the CR
energy spectrum (and thus by the radio spectral index) and also have an
imprint on the LyC escape fraction. Although other mechanisms like CR escape
due to advection/diffusion and the evolution of magnetic fields can also play
an important role. Our main results can be summarized as follows:

  1. 1.

RC spectral index between C- and S-bands
($\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$) for LzLCS sources show a large
variation from that of typical star-forming galaxies, with sources showing
very flat (¿ -0.1) to very steep spectral index (¡ -1).

  2. 2.

We find that LzLCS galaxies have a normal to significant contribution of the
non-thermal component to the total radio emission. This provides the first
direct evidence of SN activity in these galaxies and suggests that SN can
contribute to the escape of LyC photons.

  3. 3.

We find a positive correlation between
$\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$ and the escape fraction of Lyman
continuum photons, $f_{\mathrm{esc}}^{\mathrm{LyC}}$, such that strong leakers
have flat $\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$ and non-leakers have very
steep $\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$.

  4. 4.

We argue that a combination of young ages (high fth), FFA with a turnover at
GHz frequencies, and CRs with a flat spectrum (young) can lead to the
flattening of $\alpha^{\mathrm{3GHz}}_{\mathrm{6GHz}}$. Such physical
conditions which are typical of young and dense starbursts are also when
galaxies are expected to show the highest $f_{\mathrm{esc}}^{\mathrm{LyC}}$.

  5. 5.

Steep spectrum sources show evidence of declining SFR on short timescales and
are also associated with low $f_{\mathrm{esc}}^{\mathrm{LyC}}$. CR escape due
to advection/diffusion can also play a role. However, we cannot distinguish
between these processes from the current data.

  6. 6.

Our work highlights the evolutionary nature of
$f_{\mathrm{esc}}^{\mathrm{LyC}}$, on short timescales ($\sim 10$ Myrs,) and
its association with SN, CR energy spectrum, and magnetic fields in the escape
of LyC photons.

  7. 7.

The RC-SFR relation of LzLCS sources show a large scatter. Using RC as an SFR
tracer for such galaxies is difficult using a narrow frequency range.
Information on the radio-SED will be crucial to correctly calibrate the radio-
based SFRs in such galaxies. Since these galaxies are analogues of high-$z$
galaxies ($z>3$), our study gives important insights into such galaxies.
Particularly when the upcoming highly sensitive radio facilities like the SKA
and ngVLA will detect star-forming galaxies at high redshifts.

In our current study with just the C- and S-band data, we cannot robustly
estimate some of the physical parameters. In the future, we plan to conduct
low-frequency (¡ 1.5 GHz) and high-frequency (¿ 8 GHz) follow-up observations
of LzLCS sources to better study their radio-SED. This will allow us to fit
complex radio-SED models which will help us in accurately estimating the non-
thermal spectral index and the thermal fraction. This will help us in
estimating some of the physical parameters, for example, the SN rate, CR
energy spectrum, and magnetic fields and their relation to LyC escape. This
will improve our understanding of the poorly understood role of SN feedback in
the escape of LyC photons. Finally, our current radio follow up of LzLCS lacks
the presence of many strong leakers which will be followed up in the future.

###### Acknowledgements.

O. Bait (OB) is supported by the AstroSignals Sinergia Project funded by the
Swiss National Science Foundation. OB would like to thank Jennifer Schober and
Aayush Saxena for useful comments on the manuscript. OB would also like to
thank Drew Medlin and the NRAO science helpdesk team for their extensive help
with running the VLA pipeline and with the VLA data reduction. We would like
to thank Brad Koplitz for his help with setting up some of the VLA scheduling
blocks. ASL acknowledges support from the Swiss National Science Foundation.
Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P.
Sloan Foundation, the U.S. Department of Energy Office of Science, and the
Participating Institutions. SDSS-IV acknowledges support and resources from
the Center for High Performance Computing at the University of Utah. The SDSS
website is www.sdss4.org. SDSS-IV is managed by the Astrophysical Research
Consortium for the Participating Institutions of the SDSS Collaboration
including the Brazilian Participation Group, the Carnegie Institution for
Science, Carnegie Mellon University, Center for Astrophysics — Harvard &
Smithsonian, the Chilean Participation Group, the French Participation Group,
Instituto de Astrofísica de Canarias, The Johns Hopkins University, Kavli
Institute for the Physics and Mathematics of the Universe (IPMU) / University
of Tokyo, the Korean Participation Group, Lawrence Berkeley National
Laboratory, Leibniz Institut für Astrophysik Potsdam (AIP), Max-Planck-
Institut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für Astrophysik
(MPA Garching), Max-Planck-Institut für Extraterrestrische Physik (MPE),
National Astronomical Observatories of China, New Mexico State University, New
York University, University of Notre Dame, Observatário Nacional / MCTI, The
Ohio State University, Pennsylvania State University, Shanghai Astronomical
Observatory, United Kingdom Participation Group, Universidad Nacional Autónoma
de México, University of Arizona, University of Colorado Boulder, University
of Oxford, University of Portsmouth, University of Utah, University of
Virginia, University of Washington, University of Wisconsin, Vanderbilt
University, and Yale University.

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##  Appendix A Observation details of VLA S-, C- and L-band sources in our
sample.

In Table 4 we describe the details of the C- and S-band observations for our
sources. And Table 5 shows the details of the L-band observations for a sub-
sample of these sources.

Table 4: VLA C- and S-band observation details. Object | Obs. Date | Flux Cal | Phase Cal | Phase Center Ra | Phase Center Dec
---|---|---|---|---|---
J004743+015440 | 2021-11-17 | 3C 147 | J0059+0006 | 00h47m42.8s | +01d54m39.9s
J011309+000223 | 2021-12-08 | 3C 147 | J0059+0006 | 01h13m08.9s | +00d02m23.4s
J012910+145935 | 2021-10-07 | 3C 147 | J0204+1514 | 01h29m10.2s | +14d59m34.7s
J072326+414608 | 2021-11-05 | 3C 147 | J0713+4349 | 07h23m25.6s | +41d46m07.8s
J081112+414146 | 2022-02-03 | 3C 147 | J0818+4222 | 08h11m12s | +41d41m45.9s
J081409+211459 | 2021-12-01 | 3C 147 | J0842+1835 | 08h14m08.8s | +21d14m59.2s
J082652+182052 | 2022-01-02 | 3C 147 | J0842+1835 | 08h26m51.8s | +18d20m51.8s
J083440+480541 | 2022-02-15 | 3C 147 | J0818+4222 | 08h34m40.1s | +48d05m40.9s
J090918+392925 | 2022-02-15 | 3C 147 | J0818+4222 | 09h09m18.4s | +39d29m24.7s
J091113+183108 | 2021-12-30 | 3C 147 | J0854+2006 | 09h11m13.3s | +18d31m08.2s
J091207+523960 | 2022-02-08 | 3C 147 | J0921+6215 | 09h12m07.2s | +52d39m59.9s
J091208+505009 | 2021-11-25 | 3C 147 | J0834+5534 (S), J0832+4913 (C) | 09h12m08.1s | +50d50m08.6s
J091703+315221 | 2022-01-21 | 3C 147 | J0956+2515 | 09h17m02.5s | +31d52m20.6s
J092552+395714 | 2022-02-14 | 3C 147 | J0920+4441 | 09h25m51.9s | +39d57m13.7s
J094001+593244 | 2022-01-27 | 3C 147 | J0921+6215 | 09h40m00.7s | +59d32m44.4s
J095236+405249 | 2022-01-12 | 3C 147 | J0920+4441 | 09h52m35.9s | +40d52m48.8s
J095700+235709 | 2021-11-06 | 3C 147 | J0956+2515 | 09h57m00.4s | +23d57m09s
J095838+202508 | 2022-01-06 | 3C 147 | J0956+2515 | 09h58m38.4s | +20d25m07.6s
J101401+523251 | 2022-01-22 | 3C 286 | J1035+5628 | 10h14m01.5s | +52d32m50.7s
J102615+633308 | 2022-02-28 | 3C 286 | J1035+5628 | 10h26m15.2s | +63d33m08.5s
J103344+635317 | 2022-03-01 | 3C 286 | J1035+5628 | 10h33m44.1s | +63d53m17.2s
J103816+452718 | 2022-02-19 | 3C 286 | J0948+4039 | 10h38m16.2s | +45d27m17.8s
J105117+474357 | 2022-02-19 | 3C 286 | J1035+5628 | 10h51m16.9s | +47d43m57s
J105331+523753 | 2022-02-28 | 3C 286 | J1035+5628 | 10h53m30.8s | +52d37m52.9s
J110452+475204 | 2022-03-03 | 3C 286 | J1035+5628 | 11h04m52.2s | +47d52m04.3s
J112224+412052 | 2022-02-17 | 3C 286 | J1146+3958 | 11h22m24.3s | +41d20m52.4s
J112848+524509 | 2022-02-20 | 3C 286 | J1035+5628 | 11h28m47.6s | +52d45m09.4s
J112933+493525 | 2022-02-22 | 3C 286 | J1146+3958 | 11h29m32.8s | +49d35m24.6s
J113304+651341 | 2022-02-20 | 3C 286 | J1313+6735 | 11h33m03.8s | +65d13m41.4s
J115855+312559 | 2022-03-02 | 3C 286 | J1146+3958 | 11h58m54.8s | +31d25m59.2s
J115959+382422 | 2022-02-27 | 3C 286 | J1146+3958 | 11h59m58.9s | +38d24m22s
J120934+305326 | 2022-03-03 | 3C 286 | J1215+3448 | 12h09m34.5s | +30d53m26.2s
J121915+453930 | 2022-02-22 | 3C 286 | J1146+3958 | 12h19m14.7s | +45d39m29.6s
J123519+063556 | 2022-02-22 | 3C 286 | J1254+1141 | 12h35m18.5s | +06d35m56.4s
J124423+021540 | 2021-11-15 | 3C 286 | J1246-0730 | 12h44m23.4s | +02d15m40.5s
J124619+444902 | 2022-02-15 | 3C 286 | J1227+3635 | 12h46m19.5s | +44d49m02.4s
J124835+123403 | 2022-03-03 | 3C 286 | J1254+1141 | 12h48m34.6s | +12d34m02.9s
J124911+464535 | 2022-02-21 | 3C 286 | J1219+4829 | 12h49m11.2s | +46d45m34.5s
J125718+410221 | 2021-11-19 | 3C 286 | J1310+3220 | 12h57m18.3s | +41d02m21.4s
J130128+510451 | 2022-02-21 | 3C 286 | J1219+4829 | 13h01m28.3s | +51d04m51.2s
J131037+214817 | 2022-02-27 | 3C 286 | J1327+2210 | 13h10m36.7s | +21d48m17s
J131419+104739 | 2022-02-27 | 3C 286 | J1347+1217 | 13h14m18.9s | +10d47m39s
J131904+510309 | 2021-12-16 | 3C 286 | J1219+4829 | 13h19m03.8s | +51d03m09.3s
J132633+421824 | 2022-02-21 | 3C 286 | J1310+3220 | 13h26m33.4s | +42d18m24.1s
J132937+573315 | 2022-02-28 | 3C 286 | J1400+6210 | 13h29m37.4s | +57d33m15.2s
J134559+112848 | 2022-03-02 | 3C 286 | J1347+1217 | 13h45m58.9s | +11d28m47.8s
J141013+434435 | 2021-12-27 | 3C 286 | J1416+3444 | 14h10m13s | +43d44m35.1s
J144010+461937 | 2022-02-21 | 3C 286 | J1349+5341 | 14h40m09.9s | +46d19m37s
J151707+370512 | 2022-02-22 | 3C 286 | J1602+3326 | 15h17m07.4s | +37d05m12.3s
J155945+403325 | 2022-02-20 | 3C 286 | J1613+3412 | 15h59m44.6s | +40d33m25.3s
J160437+081959 | 2022-03-03 | 3C 286 | J1557-0001 | 16h04m36.7s | +08d19m59.1s
J164607+313054 | 2022-02-22 | 3C 286 | J1613+3412 | 16h46m06.5s | +31d30m53.5s
J164849+495751 | 2021-12-24 | 3C 286 | J1739+4737 | 16h48m49.4s | +49d57m50.9s
Table 5: VLA L-band observation details. Object | Obs. Date | Flux Cal | Phase Cal | Phase Center Ra | Phase Center Dec
---|---|---|---|---|---
J004743+015440 | 2022-01-28 | 3C 147 | J0059+0006 | 00h47m42.8s | +01d54m39.9s
J011309+000223 | 2022-01-29 | 3C 147 | J0059+0006 | 01h13m08.9s | +00d02m23.4s
J012910+145935 | 2022-01-17 | 3C 147 | J0204+1514 | 01h29m10.2s | +14d59m34.7s
J072326+414608 | 2022-01-28 | 3C 147 | J0713+4349 | 07h23m25.6s | +41d46m07.8s
J081112+414146 | 2022-02-21 | 3C 147 | J0818+4222 | 08h11m12s | +41d41m45.9s
J081409+211459 | 2022-01-28 | 3C 147 | J0842+1835 | 08h14m08.8s | +21d14m59.2s
J082652+182052 | 2022-02-21 | 3C 147 | J0842+1835 | 08h26m51.8s | +18d20m51.8s
J083440+480541 | 2022-01-19 | 3C 147 | J0818+4222 | 08h34m40.1s | +48d05m40.9s
J090918+392925 | 2022-02-22 | 3C 147 | J0818+4222 | 09h09m18.4s | +39d29m24.7s
J091113+183108 | 2022-01-30 | 3C 147 | J0854+2006 | 09h11m13.3s | +18d31m08.2s
J091207+523960 | 2022-01-21 | 3C 147 | J0921+6215 | 09h12m07.2s | +52d39m59.9s
J091208+505009 | 2022-01-24 | 3C 147 | J0834+5534 | 09h12m08.1s | +50d50m08.6s
J091703+315221 | 2022-01-30 | 3C 147 | J0956+2515 | 09h17m02.5s | +31d52m20.6s
J092552+395714 | 2022-01-25 | 3C 147 | J0920+4441 | 09h25m51.9s | +39d57m13.7s
J095236+405249 | 2022-02-18 | 3C 147 | J0920+4441 | 09h52m35.9s | +40d52m48.8s
J095700+235709 | 2022-02-27 | 3C 147 | J0956+2515 | 09h57m00.4s | +23d57m09s
J095838+202508 | 2022-02-28 | 3C 147 | J0956+2515 | 09h58m38.4s | +20d25m07.6s

##  Appendix B VLA S- and C-band images for the detections in our sample.

Figure 15:  Shows the RC images for all the LzLCS detections in S-band (blue
contours) and C-band (orange contours). The black crosshair represents the
optical centre. The synthesized beam is shown blue (orange) filled ellipticals
for S-band (C-band). The contours start from 3-$\sigma$, and successively
increase in powers of $\sqrt{2}$. The dashed contours represent the
-3-$\sigma$ contour.

Figure 16:  Continued from Fig. 15.

In Fig. 15 and 16 we show the VLA C-band (in orange) and S-band (in blue)
contours overlaid on each other for our detection sample.

##  Appendix C Radio-SED for our LzLCS sample at GHz frequencies.

In Fig. 17 and 18 we present the radio-SED across C-, S- and also L-band (for
those with C, S-band detections and L-band observations.).

Figure 17:  Same as Fig. 3 for the remaining LzLCS sample with C- and S-bands
detections. The red arrows represent the 3-$\sigma$ upper limits for L-band
non-detections in our sample.

Figure 18: Continued from Fig. 17.