2.1. GLEAM Exgal and SGP catalogues and images (76–227 MHz)

We use the GLEAM Extragalactic (Exgal; Hurley-Walker et al., Reference Hurley-Walker2017) and South Galactic Pole (SGP; Franzen et al., Reference Franzen, Hurley-Walker, White, Hancock, Seymour, Kapińska, Staveley-Smith and Wayth2021) data releases as our low-frequency basis data. GLEAM Exgal is based on the first year (2013–2014) of GLEAM observations. It covers 24 831 $\mathrm{\,deg}^2$ at Dec $< +30^{\circ} $ , excluding the strip at Galactic latitude $|b| < 10^{\circ} $ and a few regions around bright, complex sources such as the Magellanic Clouds. The GLEAM Exgal catalogue contains 307 455 source components with 20 separate flux density measurements between 76 and 227 MHz selected from a wide-band image centred at 200 MHz, with an angular resolution of ${\approx}2$ arcmin. The typical rms noise in the wide-band image is ${\approx}10$ mJy/beam. Spectral indices between 76 and 227 MHz were derived by fitting a power law to the sub-band flux densities.

GLEAM SGP is based on a subset of both years (2013–2015) of GLEAM observations. It covers an area of 5 113 $\mathrm{\,deg}^2$ surrounding the South Galactic Pole at $20^{\mathrm{h}} 40^{\mathrm{m}} < \mathrm{RA} < 05^{\mathrm{h}} 04^{\mathrm{m}}$ and $-48^{\circ} < \mathrm{Dec} < -2^{\circ} $ . The GLEAM SGP catalogue contains 108 851 source components with 20 separate flux density measurements between 76 and 227 MHz selected from a wide-band image centred at 216 MHz, with an angular resolution of ${\approx}2$ arcmin. The typical rms noise in the wide-band image is ${\approx}5$ mJy/beam. This is still well above the classical confusion limit of ${\approx}2$ mJy/beam (Franzen et al., Reference Franzen, Vernstrom, Jackson, Hurley-Walker, Ekers, Heald, Seymour and White2019).

Spectral indices across the GLEAM band were derived by fitting a power law to the sub-band flux densities. The catalogue also contains integrated flux densities at 200 MHz, $S_{200\,\textrm{MHz}}$ , as in the GLEAM Exgal catalogue. If the GLEAM intra-band spectrum could be well fitted by a power law, $S_{200\,\textrm{MHz}}$ was derived from the power-law fit. Otherwise, $S_{200\,\textrm{MHz}}$ was derived by extrapolating the 216-MHz integrated flux density from the wide-band image to 200 MHz assuming $\alpha = -0.8$ .

We refer to the area of sky covered by the GLEAM SGP catalogue, which is depicted by the red shading in Figure 1, as the ‘deep’ region. We refer to the area of sky covered by the GLEAM Exgal catalogue at $\mathrm{Dec} < 0^{\circ} $ and outside the deep region as the ‘shallow’ region. The shallow region is depicted by the cyan shading in Figure 1. The deep and shallow regions are fully described in Table 2. In constructing the GLEAM-6dFGS sample, we use the GLEAM SGP catalogue in the deep region and the GLEAM Exgal catalogue in the shallow region.

We use cutouts from the GLEAM Exgal and SGP wide-band images for visual inspection of the cross-identifications outlined in Section 3.4.

2.2. SUMSS catalogue and images (843 MHz)

SUMSS covers the entire extragalactic sky ( $|b|>10^\circ$ ) south of Dec $-30^{\circ} $ at 843 MHz. It has a limiting peak brightness of 6 mJy/beam at $\mathrm{Dec} \leq -50^{\circ} $ and 10 mJy/beam at $\mathrm{Dec} > -50^{\circ} $ . The angular resolution is $45\times45 \operatorname{cosec} |\mathrm{Dec}|\,\mathrm{arcsec}^{2}$ .

We use cutouts from SUMSS for the visual inspection at $\mathrm{Dec} < -39.5^{\circ} $ . If necessary, we use the SUMSS catalogue to correct the SUMSS flux density quoted in the NVSS/SUMSS-6dFGS catalogue described in Section 2.4.

2.3. NVSS catalogue and images (1.4 GHz)

NVSS covers the entire sky north of Dec $-40^{\circ} $ at 1.4 GHz. It has a 5- $\sigma$ limit in peak brightness of ${\approx}2.5$ mJy/beam and an angular resolution of 45 arcsec.

We use cutouts from NVSS for the visual inspection at $\mathrm{Dec} \geq -39.5^{\circ} $ . If necessary, we use the NVSS catalogue to correct the NVSS flux density quoted in the NVSS/SUMSS-6dFGS catalogue.

2.4. The 6dFGS and the NVSS/SUMSS-6dFGS catalogue

2.4.2. The NVSS/SUMSS-6dFGS catalogue

As noted by White et al. (Reference White2020a), many GLEAM sources have complex structure at low frequencies. Combined with the relatively low spatial resolution of the GLEAM images ( ${\sim}2{-}3$ arcmin synthesised beam), this makes it challenging to identify the host galaxies of GLEAM sources in an automated way from low-frequency data alone. We, therefore, chose to use higher-frequency radio data from NVSS at 1.4 GHz and SUMSS at 843 MHz, which have higher spatial resolution than GLEAM, as a guide for cross-matching the 6dFGS and GLEAM catalogues as well as a source of radio spectral classifications for the GLEAM-6dFGS radio-matched galaxies.

Mauch & Sadler (Reference Mauch and Sadler2007) used data from NVSS to identify 7 824 radio sources with galaxies brighter than K = 12.75 mag in the Second Incremental Data Release (DR2) of the 6dFGS. These authors later used the third and final 6dFGS data release (DR3) to compile a (currently unpublished) catalogue of NVSS-6dFGS and SUMSS-6dFGS cross-identifications at $\mathrm{Dec} \geq -39.5^{\circ} $ and $\mathrm{Dec} < -39.5^{\circ} $ , respectively, which was used for this study.

Table 2. Description of the deep and shallow regions of GLEAM used in the analysis of this paper

^† The peeled sources are Hydra A, Pictor A, Hercules A, Virgo A, Tau A, Cygnus A and Cassiopeia A; their positions are listed in Hurley-Walker et al. (Reference Hurley-Walker2017).

The NVSS/SUMSS-6dFGS catalogue was generated using the same methodology described by Mauch & Sadler (Reference Mauch and Sadler2007) and contains ${\approx}12\,500$ 6dFGS radio matches in total (representing around 10% of the 125 071 galaxies in the 6dFGS DR3 catalogue).

A common method of determining the origin (star formation or jets originating from the central supermassive black hole) of the bulk of the radio emission in nearby galaxies is through their optical emission line properties. The optical spectrum of each galaxy in the NVSS/SUMSS-6dFGS catalogue was visually classified by Tom Mauch using the same method applied by Mauch & Sadler (Reference Mauch and Sadler2007). If the spectrum showed evidence of a galaxy with ongoing star formation, with strong narrow emission lines of H $\alpha$ and H $\beta$ typical of HII regions, the source was classified as a SF galaxy. Otherwise, the source was classified as an AGN. The AGN class was further divided into three subclasses: (i) sources with pure absorption line spectra, classed as ‘Aa’; (ii) sources with both absorption lines and weak, narrow emission lines characteristic of Low-Ionisation Nuclear Emission-line Regions (LINERs), classed as ‘Aae’; and (iii) sources with strong nebular emission lines (stronger than any hydrogen Balmer emission lines), classed as ‘Ae’.

It is usually straightforward to tell the dominant physical process responsible for the radio emission from the optical spectra (Sadler et al., Reference Sadler, McIntyre, Jackson and Cannon1999; Jackson & Londish, Reference Jackson and Londish2000). The only significant complication is with the emission line AGN (Ae and Aae objects) that make up 14% of the GLEAM-6dFGS sample. These are a heterogeneous class of objects where both an AGN and star formation may contribute significantly to the radio emission, the dominant mechanism varying between sources (e.g. Ching et al., Reference Ching2017). It is likely that some of the weaker radio sources in the Ae/Aae class arise mainly from star formation processes. We return to this point when deriving the local RLF for radio AGN and SF galaxies in Section 5.2.

The NVSS/SUMSS-6dFGS catalogue includes the target name of the object from the 6dFGS data base; the position of the object and its total infrared K-band magnitude from the Two Micron All-Sky Survey Extended Source Catalogue (2MASS XSC; Jarrett et al., Reference Jarrett, Chester, Cutri, Schneider, Rosenberg, Huchra and Mader2000), as listed in the 6dFGS database; the optical redshift and redshift quality flag, as listed in the 6dFGS catalogue; the spectral classification for galaxies with a good-quality 6dFGS spectrum from the visual inspection; and the integrated flux density of the radio counterpart in NVSS or SUMSS.

Although NVSS and SUMSS flux densities are included in the NVSS/SUMSS-6dFGS catalogue, all the flux densities used in this paper were checked and re-derived from the original NVSS and SUMSS catalogues.

2.5. The AllWISE catalogue in the mid-infrared

The Wide-field Infrared Survey Explorer (WISE; Wright et al., Reference Wright2010) covers the whole sky at 3.4 (W1), 4.6 (W2), 12 (W3), and 22 (W4) $\mu \mathrm{m}$ . We use the AllWISE data release catalogue by Cutri et al. (Reference Cutri2013) to characterise the GLEAM-6dFGS sources in the mid-infrared. The angular resolutions are 6.1, 6.4, 6.5, and 12.0 arcsec, and the flux sensitivities at 5 $\sigma$ are 0.054, 0.071, 0.73, and 5.0 mJy/beam, in W1, W2, W3, and W4, respectively.

2.6. Other radio and optical data

The following data were also used, where available, for the visual inspection to confirm the cross-identification:

1. • Cutouts from the alternative data release 1 of the TIFR GMRT Sky Survey (TGSS ADR1; Intema et al., Reference Intema, Jagannathan, Mooley and Frail2017). The survey covers the entire sky north of Dec $-53^{\circ} $ at 150 MHz to a typical rms sensitivity of 3.5 mJy/beam. The angular resolution is $25{\times}25 \operatorname{sec}\,(\mathrm{Dec} - 19^{\circ} )\,\mathrm{arcsec}^{2}$ at $\mathrm{Dec} < +19^{\circ} $ .

2. • Cutouts from the 1.4-GHz Faint Images of the Radio Sky at Twenty Centimetres (FIRST; Becker, White, & Helfand, Reference Becker, White and Helfand1995) survey, which covers ${\approx}10\,575\,\mathrm{deg}^{2}$ of the north Galactic cap. The typical sensitivity is 0.13 mJy/beam and the angular resolution 5 arcsec.

3. • Quick Look images from the first epoch of the VLA Sky Survey (VLASS; Lacy et al., Reference Lacy2020), which covers the entire sky at $\mathrm{Dec} > -40^{\circ} $ at 2–4 GHz. The sensitivity is ${\approx}120\,\mu$ Jy/beam and the angular resolution 2.5 arcsec.

4. • Optical blue ( $B_{J}$ -band) images from the SuperCOSMOS Sky Survey (Hambly et al., Reference Hambly2001) available over the entire southern sky to a typical AB depth of $B_{J} < 21$ .

3.1. Cross-matching GLEAM with 6dFGS, NVSS, and SUMSS

We use the topcat software (Taylor, Reference Taylor2005) to cross-match the NVSS/SUMSS-6dFGS catalogue with the GLEAM SGP catalogue in the deep region and the GLEAM Exgal catalogue in the shallow region. We conduct a simulation to determine the best search radius to automatically accept GLEAM counterparts to 6dFGS galaxies. The black histogram in Figure 2 shows the number of GLEAM sources with identified 6dFGS counterparts as a function of angular separation between the GLEAM and 6dFGS positions. We generate a list of simulated sources by offsetting the GLEAM positions by 10 arcmin in Dec and cross-match the simulated catalogue with the NVSS/SUMSS-6dFGS catalogue. The red histogram shows the number of matches between the simulated catalogue and the NVSS/SUMSS-6dFGS catalogue as a function of angular separation. At a separation of 100 arcsec, the number of false matches is roughly equal to the number of real matches. Matches beyond this separation are therefore unlikely to be genuine. At a separation of 50 arcsec, roughly half the matches are false.

Figure 2. Results of a simulation to determine the best search radius to automatically accept GLEAM counterparts to 6dFGS sources. The black histogram shows the number of GLEAM sources with identified 6dFGS counterparts as a function of the angular separation between the GLEAM and 6dFGS positions. The red histogram shows the results obtained for a simulated catalogue generated by offsetting the GLEAM positions by 10 arcmin in Dec. The dashed vertical line at 50 arcsec marks the chosen search radius.

We conclude that a search radius of 50 arcsec provides a good compromise between completeness and reliability. Using this search radius, we obtain a total of 3 495 matches.

3.2. Radio completeness

We calculate the flux density levels at which the sample is close to 100% complete in the shallow and deep regions. Hurley-Walker et al. (Reference Hurley-Walker2017) created maps of the spatial variation of the completeness in GLEAM Exgal at 33 flux density levels spanning the range 25 mJy to 1 Jy. We use these maps to calculate the mean completeness at each flux density level within the shallow region. The mean completeness is found to be ${\approx}90$ % at $S_{200\,\mathrm{MHz}} = 100$ mJy. The completeness is close to uniform except in a relatively small region of sky covering about $500\,\mathrm{deg}^{2}$ at $16^{\mathrm{h}} < \mathrm{RA} < 20^{\mathrm{h}}$ and $\mathrm{Dec} < -60^{\circ} $ , where it drops well below 90%.

The GLEAM SGP catalogue is estimated to be 90% complete at $S_{216\,\mathrm{MHz}} = 50$ mJy (Franzen et al., Reference Franzen, Hurley-Walker, White, Hancock, Seymour, Kapińska, Staveley-Smith and Wayth2021). The completeness is close to uniform across the entire region covered by the catalogue. The completeness limit at 200 MHz will depend on the distribution of spectral indices between 200 and 216 MHz. In estimating the completeness limit at 200 MHz, we take a conservative approach and assume that all sources have a spectral index of $-1.2$ . Thus, we set the completeness limit at 200 MHz to 55 mJy.

We note that genuine GLEAM-6dFGS matches which are too faint to appear in NVSS/SUMSS will be omitted from our sample. NVSS is close to 100% complete at $S_{1\,400\,\mathrm{MHz}} = 3.4$ mJy. SUMSS has a completeness limit of $S_{843\,\mathrm{MHz}} = 10$ mJy at $\mathrm{Dec} \leq -50^{\circ} $ and of $S_{843\,\mathrm{MHz}} = 18$ mJy at $\mathrm{Dec} > -50^{\circ} $ . Table 3 shows the limiting spectral index between 200 and 1 400/843 MHz in five different regions of the GLEAM-6dFGS sample depending on the survey data available. The limiting spectral indices assume that the angular size of the source is not significantly larger than 45 arcsec, the angular resolution of NVSS and SUMSS.

Table 3. The limiting spectral indices, $\alpha_{\mathrm{lim}}$ , between 200 and 1 400/843 MHz in five different regions of the GLEAM-6dFGS sample depending on the survey data available. Sources with $\alpha < \alpha_{\mathrm{lim}}$ may not appear in the sample. The final number of GLEAM-6dFGS sources in each region is given in the last column.

The highest limiting spectral index is –0.78 in the section of the deep region covered by SUMSS. The fraction of sources with spectral indices between 200 MHz and ${\sim}1$ GHz, $\alpha_{\mathrm{high}}$ , steeper than –0.78 is ${\approx}21\%$ based on the distribution of $\alpha_{\mathrm{high}}$ for the GLEAM-6dFGS sources (see Section 4.3). We therefore expect ${\approx}21\%$ of genuine GLEAM-6dFGS associations to be missing from the sample in this region close to the 55 mJy flux density cut, assuming that the source angular size is ${\lesssim}45$ arcsec. However, this region only represents 5% of the total area covered by the sample.

The limiting spectral indices in the other regions listed in Table 3 lie between $-1.19$ and $-1.74$ . We can obtain an estimate of the fraction of sources with $\alpha < -1.2$ from Seymour et al., in preparation, who cross-matched GLEAM SGP with the third data release from the Australia Telescope Large Area Survey (ATLAS DR3; Franzen et al., Reference Franzen2015); ATLAS DR3 covers the Chandra Deep Field South (CDFS; Giacconi et al., Reference Giacconi2001), and the European Large Area ISO Survey - South 1 (ELAIS-S1; Oliver et al., Reference Oliver2000) to an rms depth of ${\approx}15\,\mu \mathrm{Jy/beam}$ at 1.4 GHz. Of the 134 GLEAM SGP sources lying in CDFS and ELAIS-S1, only four (3%) were found to have $\alpha < -1.2$ . From these spectral index limits, we determine that the GLEAM-6dFGS sample is biased against diffuse radio sources with very steep spectra such as radio haloes and relics. Such sources may be detected by directly cross-matching the GLEAM catalogue with the 6dFGS catalogue, but this is beyond the scope of this paper. We do not expect any bias against this source class to significantly affect the overall completeness of the sample and the measured local RLF.

3.3. Source filtering

The determination of the local RLF requires a sample which is complete to the limits of the radio and optical surveys from which it is derived. As described in Section 3.2, the shallow region is complete to $S_{200\,\mathrm{MHz}} = 100$ mJy and the deep region to $S_{200\,\mathrm{MHz}} = 55$ mJy. In order to form a sample which is complete in the radio, we discard all sources with $S_{200\,\mathrm{MHz}} < 100$ mJy and $S_{200\,\mathrm{MHz}} < 55$ mJy in the shallow and deep regions, respectively. Since the 6dFGS DR3 catalogue is close to 100% complete at $K < 12.65$ mag (Jones et al., Reference Jones2009), we discard sources with $K > 12.65$ mag. Finally, we remove sources with unreliable redshifts (redshift quality 1 or 2), those associated with Galactic stars (redshift quality 6) and those with $z < 0.001$ , leaving a total of 1 688 sources.

3.4. Visual inspection

For all 1 688 selected sources, we visually inspect optical J-band images of size 15 arcmin from the SuperCOSMOS Sky Survey overlaid with radio contours from GLEAM and NVSS/SUMSS. Firstly, we identify and remove unreliable GLEAM-6dFGS matches. Secondly, we identify which NVSS/SUMSS components are associated with each source and, if necessary, correct the total NVSS/SUMSS flux density. Thirdly, we use the overlays to identify sources whose flux densities in GLEAM are significantly affected by confusion from adjacent sources and apply a deblending method to correct the GLEAM flux densities.

3.4.1. Removing false GLEAM-6dFGS matches

The probability of a GLEAM-6dFGS match being genuine is dependent on the GLEAM-6dFGS offset. The simulations used to determine the best search radius to automatically accept GLEAM-6dFGS identifications (see Section 3.1) indicate that the reliability of the cross-matches is ${\approx}99\%$ for position offsets less than 30 arcsec and ${\approx}73\%$ for position offsets between 30 and 50 arcsec.

We visually inspect the overlays of all sources, paying particular attention to sources with position offsets larger than 30 arcsec and identify 22 false GLEAM-6dFGS matches. These sources are subsequently removed from the sample. The false GLEAM-6dFGS matches are typically the result of confusion with an adjacent source, displacing the position of the GLEAM component away from the 6dFGS position and boosting its flux density; an example of such a case is shown in Figure 3.

Figure 3. Example overlay for a source with a relatively large GLEAM-6dFGS position offset (47.1 arcsec). Radio contours from GLEAM (200 MHz; red) and NVSS (1 400 MHz; blue) are overlaid on the SuperCOSMOS J-band image (inverted greyscale). For each set of contours, the lowest contour is at the 3 $\sigma$ level, where $\sigma$ is the local rms, with the number of $\sigma$ increasing by a factor of $\sqrt{2}$ with each subsequent contour. Catalogue positions from GLEAM (magenta plus signs) and NVSS (magenta crosses) are plotted. The cyan triangle shows the optical position of the source in 6dFGS. The large GLEAM-6dFGS position offset is the result of confusion with an adjacent source. The GLEAM-6dFGS match is not genuine and the source is removed from the sample.

3.4.2. Identifying multi-component sources

We attempt to automatically identify multi-component sources in NVSS/SUMSS using the NVSS/SUMSS cutouts. The integration area of a source is taken to consist of all pixels in the NVSS/SUMSS image that are within the contiguous 3- $\sigma$ contour level, bounding the GLEAM position in question. All catalogued NVSS/SUMSS components located within the integration area are considered to be associated with the GLEAM component. We calculate the brightness-weighted centroid of the NVSS/SUMSS emission from the positions and flux densities of the associated NVSS/SUMSS components. The errors on the positions of the individual NVSS/SUMSS components are assumed to be correlated when calculating the centroid’s position error. We also calculate the total flux density of the individual NVSS/SUMSS components. Again, we take a conservative approach and assume that the component flux density errors are correlated.

This automated technique can fail to identify the correct NVSS/SUMSS components in the following two situations: (i) the NVSS/SUMSS components of a very extended source are well separated and there is no extended emission $>\!3\sigma$ linking the components; (ii) unrelated point sources lie sufficiently close together on the sky such that they are located within the same integration area. In order to identify and correct these errors, we visually inspect all the overlays. Figure 4 shows example overlays for an extended AGN and SF galaxy where the NVSS/SUMSS components are correctly identified using the automated procedure.

Figure 4. Example overlays for an extended FRII at $z = 0.0706$ (GLEAM J010150-025226; left) and a face-on spiral galaxy at $z = 0.0084$ (GLEAM J102902-443916; right) in the GLEAM-6dFGS sample. Radio contours from GLEAM (200 MHz; red) and NVSS/SUMSS (1 400/843 MHz; blue) are overlaid on the SuperCOSMOS J-band image (inverted greyscale). For each set of contours, the lowest contour is at the 3 $\sigma$ level, where $\sigma$ is the local rms, with the number of $\sigma$ increasing by a factor of $\sqrt{2}$ with each subsequent contour. Catalogue positions from GLEAM (magenta plus signs) and NVSS or SUMSS (magenta crosses) are plotted. The brightness-weighted centroid position, calculated using the NVSS/SUMSS components, is indicated by a cyan circle. The cyan triangle shows the optical position of the source in 6dFGS.

Where necessary, we use higher-resolution radio data from FIRST, VLASS, and TGSS to help identify by eye the correct NVSS/SUMSS components. The VLASS Quick Look images can be particularly useful to determine whether overlapping NVSS sources are physically related or not, as shown in Figure 5. In the final GLEAM-6dFGS catalogue, a total of 372 sources (23%) are associated with more than one NVSS/SUMSS component. The source (GLEAM J225615-361754) with the highest number of components is a giant radio galaxy with 10 NVSS components, an angular size of ${\approx}14$ arcmin and a linear size of ${\approx}1.4$ Mpc.

Figure 5. Example overlays including higher-resolution radio data from VLASS to help determine whether overlapping NVSS components are physically related or not. Radio contours from VLASS (2–4 GHz; orange) are plotted, with the lowest contour at the 5 $\sigma$ level and the number of $\sigma$ increasing by a factor of $\sqrt{2}$ with each subsequent contour. The other contours and symbols are as described in Figure 4. Left: the southern NVSS component is detected as an FRI in VLASS. The northern NVSS component, also detected in VLASS, is physically unrelated. Right: the western NVSS component is detected as a double radio galaxy in VLASS and is physically unrelated with the eastern NVSS component.

We identify 13 sources that are associated with more than 1 GLEAM component; the highest number of GLEAM components is four. As expected, the number of multi-component sources in GLEAM is much smaller due to the larger GLEAM beam size of ${\approx}2$ arcmin. We sum the integrated flux densities of the individual GLEAM components to obtain the total, integrated flux density at 200 MHz and at each of the 20 sub-band frequencies. When calculating the errors on the total flux densities, we assume that the component flux density errors are correlated. We provide the name(s) of any associated GLEAM component(s) in the GLEAM-6dFGS catalogue.

3.4.3. Correcting the GLEAM flux densities for confusion

From the visual inspection, we find instances where the MWA beam has blended unrelated NVSS/SUMSS sources together. We identify 254 sources whose GLEAM flux densities are likely to be significantly overestimated due to confusion and that require re-fitting. An example of such a source is shown in Figure 6.

Figure 6. Example overlay where two distinct galaxies detected in SUMSS are confused in GLEAM. The contours and symbols are as described in Figure 4.

For each of these sources, we use the priorised fitting mode of the aegean source finder (Hancock et al., Reference Hancock, Murphy, Gaensler, Hopkins and Curran2012; Hancock, Trott, & Hurley-Walker, Reference Hancock, Trott and Hurley-Walker2018) to deblend the GLEAM emission from the 6dFGS galaxy. We extract flux densities from the GLEAM wide-band and sub-band images at the positions of all catalogued NVSS/SUMSS components within 10 arcmin from the 6dFGS galaxy, simultaneously. The flux densities of the components are measured using Gaussian fitting. Only the peak flux densities of the components are allowed to vary in the Gaussian fitting; the positions of the components are fixed and their shapes are set to the synthesised beam of the GLEAM image. This is because the Gaussian fitting can be poorly constrained when allowing the positions and shapes of the components to vary. In any case, given the higher resolution of NVSS and SUMSS, the emission in GLEAM can generally be very well modelled with point sources at the positions of the NVSS and SUMSS components.

The deblended flux density of the NVSS/SUMSS component associated with the 6dFGS galaxy is then taken as our new estimate of the GLEAM flux density at each frequency. If more than one NVSS/SUMSS component is associated with the 6dFGS galaxy, we sum the component flux densities to obtain the total, integrated flux density. After applying this procedure, we obtain deblended flux densities at 20 frequencies between 76 and 227 MHz from the sub-band images. We also obtain deblended flux densities at 200 and 216 MHz from the wide-band images in the shallow and deep regions, respectively. The intra-band spectral index is re-measured by fitting a power law to the 20 sub-band flux densities. In the deep region, a best estimate of the 200-MHz deblended flux density is derived using the same method as in the original GLEAM SGP catalogue (see Section 2.1).

For about one-third of the sources that are refitted, the deblended 200-MHz flux density falls below the 200-MHz flux density limit of the GLEAM-6dFGS sample (100 mJy in the shallow region and 55 mJy in the deep region). After removing these sources, the final number of sources in the GLEAM-6dFGS sample is 1 590. In the GLEAM-6dFGS catalogue, we use the ‘refitted flag’ to indicate which sources have refitted GLEAM measurements. We also flag sources for which the GLEAM flux densities, and in some cases also the NVSS/SUMSS flux densities, are judged to be unreliable due to severe confusion (‘confusion flag’) based on the visual inspection. A total of 24 sources are flagged in this way; their flux densities are likely to be overestimated.

3.5. Cross-matching with AllWISE

We use the ‘CDS Upload X-Match’ facility of topcat to cross-match the GLEAM-6dFGS sample with the AllWISE catalogue. At least one match within 5 arcsec of the 6dFGS optical position is obtained for 1 575 of the 1 590 GLEAM-6dFGS sources. In the final GLEAM-6dFGS catalogue, we record the name, position, and mid-infrared magnitudes (and errors) of the closest AllWISE match within 5 arcsec of the 6dFGS optical position.

3.6. Final catalogue description

In Table 4, we describe the 87 columns of the GLEAM-6dFGS source catalogue. The electronic version of the catalogue is available from VizieR.

Table 4. Column numbers, names, units, descriptions and first row entries for the 87 columns in the GLEAM-6dFGS source catalogue. All reported magnitudes are in the Vega system.

We use the GLEAM and NVSS/SUMSS data to calculate two-point spectral indices:

(1) \begin{equation}\alpha_{\mathrm{high}} = \frac{\ln[ S(\nu_{1}) / S(\nu_{2}) ]} {\beta} \, ,\end{equation}

where $\nu_{1} = 200$ MHz,

(2) \begin{equation}\nu_{2} =\begin{cases} 843\,\mathrm{MHz} & \text{if Dec $< -39.5^{\circ} $} \\[4pt] 1\,400\,\mathrm{MHz} & \text{otherwise},\end{cases}\end{equation}

$S(\nu_{1})$ is the integrated flux density at $\nu_{1}$ from GLEAM, $S(\nu_{2})$ is the integrated flux density at $\nu_{2}$ from NVSS/SUMSS and $\beta = \ln(\nu_{1}/\nu_{2})$ . The error on $\alpha_{\mathrm{high}}$ is given by:

(3) \begin{equation}\Delta \alpha_{\mathrm{high}} = \frac{ \sqrt{ \left[ \frac{ \Delta S(\nu_{1}) }{ S(\nu_{1}) } \right]^{2} + \left[ \frac{ \Delta S(\nu_{2}) }{ S(\nu_{2}) } \right]^{2} } } {\beta} \, .\end{equation}

The 200-MHz flux density errors quoted in the GLEAM Exgal and SGP catalogues, $\sigma_{\mathrm{fit}}(\nu_{1})$ , only account for the image noise. We therefore set

(4) \begin{equation}\Delta S(\nu_{1}) = \sqrt{\sigma_{\mathrm{fit}}(\nu_{1})^{2} + [\epsilon S(\nu_{1})]^{2}} \, ,\end{equation}

where $\epsilon$ is the GLEAM external flux scale error. The value of $\epsilon$ is Dec-dependent but at most Decs $\epsilon = 0.08$ (see Hurley-Walker et al., Reference Hurley-Walker2017 and Franzen et al., Reference Franzen, Hurley-Walker, White, Hancock, Seymour, Kapińska, Staveley-Smith and Wayth2021 for details).

We calculate rest-frame radio luminosities at 200 MHz using a k-correction based on the spectral index, $\alpha$ , and assuming the radio emission is synchrotron emission characterised by a power law. The radio luminosity of a source with flux density $S_{\nu}$ at redshift z and luminosity distance $d_{\mathrm{L}}$ is therefore given by:

(5) \begin{equation}L_{\nu} = \frac{4 \pi d_{\mathrm{L}}^2 S_{\nu}} { (1+z)^{\alpha+1}} \, .\end{equation}

In the radio k-correction, we use the spectral index calculated from the 20 GLEAM sub-band flux densities between 76 and 227 MHz, $\alpha_{\mathrm{low}}$ , if $\Delta \alpha_{\mathrm{low}} \leq 0.2$ . This condition is met for 84% of the sources in the catalogue. For the remaining sources with $\Delta \alpha_{\mathrm{low}} > 0.2$ or no measurement of $\alpha_{\mathrm{low}}$ , we use $\alpha_{\mathrm{high}}$ in the radio k-correction.

We calculate absolute K-band magnitudes using K-band k-corrections derived by Glazebrook et al. (Reference Glazebrook, Peacock, Miller and Collins1995) from the evolutionary synthesis models of Bruzual & Charlot (Reference Bruzual and Charlot1993) assuming a delta-function burst of star formation at age 5 Gyr. The absolute K-band magnitude is given by:

(6) \begin{equation}M_{\mathrm{K,\,abs}} = M_{\mathrm{K}} - 5 \log_{10} \left(\frac{d_{\mathrm{L}}}{10\,\mathrm{pc}}\right) - K(z) \, ,\end{equation}

where $M_{\mathrm{K}}$ is the apparent K-band magnitude and K(z) is the K-band k-correction given by equation (6) of Glazebrook et al.

White et al. (Reference White2020a,b) compiled a complete sample of the ‘brightest’ radio sources ( $S_{151\,\mathrm{MHz}} > 4$ Jy) at $\mathrm{Dec} < 30^{\circ} $ from the GLEAM Exgal catalogue, the majority of which are AGN with powerful radio jets. The G4Jy sample consists of 1 863 sources and is over 10 times larger than the revised Third Cambridge Catalogue of Radio Sources (3CRR Laing, Riley, & Longair, Reference Laing, Riley and Longair1983). A total of 47 G4Jy sources are in common with the GLEAM-6dFGS sample. For reference, the final column of the catalogue (‘G4Jy_name’) gives the name of the G4Jy source. We note that not all G4Jy sources with 6dFGS hosts are expected to be included in the GLEAM-6dFGS catalogue due to the maximum allowed separation of 50 arcsec between the GLEAM and 6dFGS positions, and the source filtering applied in Section 3.3.

4.1. Flux density distribution

Figure 8 shows the 200-MHz flux density distributions of the AGN and SF galaxies in the deep and shallow regions. The number of AGN and SF galaxies is roughly equal at 100 mJy, the flux density limit in the shallow region. The sample is dominated by SF galaxies close to the flux density limit in the deep region (55 mJy).

Figure 8. Histogram of $S_{200\,\mathrm{MHz}}$ for the AGN (red) and SF galaxies (blue) in the deep (top) and shallow (bottom) regions of the GLEAM-6dFGS sample. Flux density cuts of 55 and 100 mJy are applied in the deep and shallow regions, respectively.

Figure 9. 200-MHz radio luminosity as a function of redshift for the AGN and SF galaxies in the GLEAM-6dFGS sample. The dashed line corresponds to the 200 MHz flux density limit of 55 mJy in the deep region.

4.2. Redshift and luminosity distributions

In Figure 9, we present the 200-MHz radio luminosity and redshift distributions of the AGN and SF galaxies. The AGN have a median redshift of 0.081. In comparison, the median redshift of the AGN in the NVSS-6dFGS sample by Mauch & Sadler (Reference Mauch and Sadler2007), selected at 1.4 GHz, is 0.073. The NVSS is estimated to be 99% complete at $S_{1.4\,\mathrm{GHz}} = 3.4$ mJy. The equivalent completeness limit of NVSS at 200 MHz for a compact radio source with a spectral index of –0.7 is ${\approx}13$ mJy. This is a factor of ${\approx}4{-}8$ lower than the 200-MHz flux density limits of the GLEAM-6dFGS sample. The fact that the median redshift of the AGN in the NVSS-6dFGS sample is similar to that in the GLEAM-6dFGS sample implies that AGN drop out of the GLEAM-6dFGS sample primarily due to the K-band magnitude limit.

Figure 10. Left: distribution in K-band absolute magnitude and 200-MHz radio power for the AGN in the GLEAM-6dFGS sample. The dashed line corresponds to the FRI/FRII division at 1.4 GHz from Ledlow & Owen (Reference Ledlow and Owen1996), extrapolated to K-band and 200 MHz as described in the text, with FRII galaxies lying above the line. Right: distribution in the SFR and stellar mass for the SF galaxies in the GLEAM-6dFGS sample. The stellar mass is derived from the K-band absolute magnitude and the SFR from the 200-MHz radio power and stellar mass, as described in the text. The dashed line shows the SF main sequence for galaxies in the local universe by Renzini & Peng (Reference Renzini2015) and Popesso et al. (Reference Popesso2019).

The SF galaxies form a more nearby population with a median redshift of 0.015. The median redshift (0.035) of the SF galaxies in the NVSS-6dFGS sample is more than twice as high as that in the GLEAM-6dFGS sample, indicating that it is mainly the radio flux density in GLEAM that limits the maximum distance to which SF galaxies can be detected.

As expected, there is a strong separation between the two source populations according to the radio luminosity, as seen in studies at 1.4 GHz. The AGN have median $P_{200\,\mathrm{MHz}} = 10^{24.61}\,\mathrm{W}\,\mathrm{Hz}^{-1}$ and the SF galaxies have median $P_{200\,\mathrm{MHz}} = 10^{22.79}\,\mathrm{W}\,\mathrm{Hz}^{-1}$ . Among the AGN, 82% have $P_{200\,\mathrm{MHz}} > 10^{24}\,\mathrm{W}\,\mathrm{Hz}^{-1}$ while 97% of the SF galaxies have $P_{200\,\mathrm{MHz}} < 10^{24}\,\mathrm{W}\,\mathrm{Hz}^{-1}$ . There is a strong increase in the fraction of sources with optical emission lines with decreasing luminosity: while only 19% of all AGN are classed as Aae or Ae, this fraction rises to 51% among the population of low-luminosity ( $P_{200\,\mathrm{MHz}} < 10^{24}\,\mathrm{W}\,\mathrm{Hz}^{-1}$ ) AGN overlapping in radio luminosity with the SF galaxies.

The K-band absolute magnitude as a function of the 200-MHz radio luminosity for the AGN is shown in the left panel of Figure 10. The dashed line corresponds to the 1.4-GHz FRI/FRII dividing line from Ledlow & Owen (Reference Ledlow and Owen1996), shifted to K-band and 200 MHz assuming a typical galaxy colour of $R{-}K = 3.0$ mag and a typical radio spectral index $\alpha_{200\,\textrm{MHz}}^{1\,400\,\textrm{MHz}} = -0.7$ . According to this relation, our sample of nearby AGN is overwhelmingly dominated by FRI galaxies: only 29 of the 1157 AGN (2.5%) lie above the dividing line. We note, however, that there have been a number of works showing that this division line is not nearly as clean as Ledlow & Owen suggested (see e.g. Mingo et al., Reference Mingo2019). Secondly, our sample may contain a significant fraction of compact radio sources which are often referred to as FR0 galaxies.

Table 6. Spectral index statistics for the AGN and SF galaxies with measured $\alpha_{\mathrm{low}}$ and $\alpha_{\mathrm{high}}$ in the GLEAM-6dFGS sample. $N_{\mathrm{flat}}$ is the number of flat-spectrum sources with $\alpha > -0.5$ and $N_{\mathrm{USS}}$ is the number of ultra-steep-spectrum sources with $\alpha < -1.2$ .

Table 7. Results of a linear regression analysis to investigate the dependence of $\alpha_{\mathrm{low}}$ and $\alpha_{\mathrm{high}}$ on $P_{200\,\mathrm{MHz}}$ for the AGN and SF galaxies in the GLEAM-6dFGS sample. $\rho$ is the correlation coefficient. The null hypothesis that the slope is zero is evaluated using a Wald test.

Since the low-frequency radio emission from SF galaxies is primarily due to the acceleration of electrons via shocks associated with supernovae, the radio luminosity from SF galaxies can be used as a tracer of the star formation rate (SFR; e.g. Condon, Reference Condon1992). Smith et al. (Reference Smith2021) studied the relationship between 150-MHz luminosity and SFR using LOFAR measurements at 150 MHz for a near-infrared selected sample of $z<1$ galaxies. We derive SFRs for the SF galaxies in our sample using their best-fit relation, which accounts for the stellar mass, $M_{*}$ :

(7) \begin{align}\log_{10} \left( \frac{P_{150\ \mathrm{MHz}}}{\mathrm{W} \mathrm{Hz}^{-1}} \right) = & 0.90 \log_{10} \left( \frac{SFR}{\mathrm{M}_{\odot} \mathrm{yr}^{-1}} \right)\nonumber\\& +0.33 \log_{10} \left( \frac{M_{*}} {10^{10} \mathrm{M}_{\odot}} \right) + 22.22\ .\end{align}

We extrapolate the 200-MHz radio powers measured in this paper to 150 MHz using $\alpha_{\mathrm{low}}$ , if $\Delta \alpha_{\mathrm{low}} < 0.2$ , otherwise we use $\alpha_{\mathrm{high}}$ . We derive $M_{*}$ from $M_{\mathrm{K,\,abs}}$ by combining equations (4) and (5) of Longhetti & Saracco (Reference Longhetti and Saracco2009):

(8) \begin{align}\log_{10} \frac{M_{*}} {\mathrm{M}_{\odot}} &= -0.4 \left(M_{\mathrm{K,\ abs}} - M_{\mathrm{K,\ abs,\ Sun}}\right) + \log_{10} \frac{ \Upsilon_{\mathrm{K}}} { \mathrm{M}_{\odot} / \mathrm{L}_{\odot} }\ ,\end{align}

where $\Upsilon_{\mathrm{K}}$ is the mass-to-light ratio in the K-band and $M_{\mathrm{K,\,abs,\,Sun}}$ is the absolute K-band magnitude of the Sun. Since disc/spiral galaxies in the local universe typically have $\Upsilon_{\mathrm{K}} \approx 0.6$ (McGaugh & Schombert, Reference McGaugh and Schombert2014) and $M_{\mathrm{K,\,abs,\,Sun}} \approx 3.27$ mag in the Vega system (Willmer, Reference Willmer2018):

(9) \begin{equation}\log_{10} \frac{M_{*}} {\mathrm{M}_{\odot}} \approx 1.09-0.4 M_{\mathrm{K,\,abs}}\,.\end{equation}

The SFR as a function of $M_{*}$ is plotted in the right panel of Figure 10. The dashed line shows the main sequence of SF galaxies defined by Renzini & Peng (Reference Renzini2015) at $M_{*} < 10^{10}\,\mathrm{M}_{\odot}$ and by Popesso et al. (Reference Popesso2019) at $M_{*} \geq 10^{10}\,\mathrm{M}_{\odot}$ . The SF galaxies in the GLEAM-6dFGS sample scatter around and above the high-mass end of the SF main sequence. This is consistent with them lying mainly in the ‘starburst’ and ‘LIRG/ULIRG’ regions of the WISE colour–colour diagram shown in Figure 7.

4.3. Radio spectra

All sources in the GLEAM-6dFGS catalogue have a measurement of $\alpha_{\mathrm{high}}$ , while 92% of the sources have a measurement of $\alpha_{\mathrm{low}}$ . Figure 11 illustrates the wide variety of radio SEDs in the sample.

Figure 11. Example SEDs using the 20 GLEAM sub-band flux densities between 76 and 227 MHz, the GLEAM wide-band flux density at 200 MHz and the NVSS/SUMSS flux density at 1 400/843 MHz. For each source, the solid line shows the power-law fit to the GLEAM sub-band flux densities and the dashed line connects the GLEAM wide-band flux density to the NVSS/SUMSS flux density. Yellow: AGN classed as Aae with a steep power-law spectrum across the GLEAM band; black: AGN classed as Aa peaking at around 200 MHz; red: AGN classed as Aae with a highly inverted spectrum; magenta: AGN classed as Aa with a flattening spectrum at high frequency; cyan: SF galaxy with a turnover frequency around 100 MHz; blue: SF galaxy close to the 200-MHz detection limit of the sample with a typical SED.

For $\alpha_{\mathrm{low}}$ to be quoted in the catalogue, the source must have a positive flux density in each of the 20 GLEAM sub-bands, which is not always the case at low signal-to-noise ratio (SNR). Of the 1 590 sources in the GLEAM-6dFGS sample, we investigate the spectral properties of the sub-sample of 1 439 sources that have measured $\alpha_{\mathrm{low}}$ and $\alpha_{\mathrm{high}}$ , excluding the 24 sources severely affected by confusion in GLEAM and the six sources with ‘unknown’ optical spectra. We note that this sub-sample is biased against faint sources with flat spectra across the GLEAM frequency range: the negative flux densities mostly occur in the lowest frequency sub-bands which have higher rms noise. It is also biased against diffuse radio sources with ultra-steep radio spectra, as discussed in Section 3.2. The median errors on $\alpha_{\mathrm{low}}$ and $\alpha_{\mathrm{high}}$ are 0.08 and 0.06, respectively.

The left panel of Figure 12 shows a radio colour–colour plot for all AGN in the sub-sample, which are colour-coded according to their optical spectral classification. The right panel shows a radio colour–colour plot for all SF galaxies in the sub-sample. In Table 6, the median and standard deviation values of $\alpha_{\mathrm{low}}$ and $\alpha_{\mathrm{high}}$ are provided for each class of optical spectrum. Table 6 also gives the fractions of sources with $\alpha_{\mathrm{low}} > -0.5$ , $\alpha_{\mathrm{high}} > -0.5$ , $\alpha_{\mathrm{low}} < -1.2$ and $\alpha_{\mathrm{high}} < -1.2$ .

4.3.1. AGN

For the AGN, the median value of $\alpha_{\mathrm{low}}$ is $-0.704 \pm 0.011$ and the median value of $\alpha_{\mathrm{high}}$ is $-0.600 \pm 0.010$ . The slight spectral flattening with frequency is likely due to the radio emission becoming increasingly core-dominated at high frequency. In their LoTSS-SDSS sample of nearby galaxies, Sabater et al. (Reference Sabater2019) measured the 150–1 400 MHz spectral indices of 496 radio AGN detected in NVSS and FIRST. For sources with $S_{1\,400\,\mathrm{MHz}} > 20$ mJy, they obtained a median spectral index of –0.63, which is close to the median value of $\alpha_{\mathrm{high}}$ for the AGN in the GLEAM-6dFGS sample. The comparison is performed over a similar flux density range since ${\approx}90$ % of the AGN in the GLEAM-6dFGS sample have NVSS/SUMSS flux densities above 20 mJy.

For the G4Jy sample, which extends to much higher redshifts and luminosities than the GLEAM-6dFGS sample, the spectral indices over the same frequency ranges are steeper by ${\sim}0.15$ : the median value of $\alpha$ is $-0.740 \pm 0.012$ between 151 and 843 MHz, $-0.786 \pm 0.006$ between 151 and 1 400 MHz, and $-0.829 \pm 0.006$ within the GLEAM band (White et al. Reference White2020a). In Figure 13, we investigate the dependence of $\alpha_{\mathrm{low}}$ and $\alpha_{\mathrm{high}}$ on $P_{200\,\mathrm{MHz}}$ within the GLEAM-6dFGS sample, separating between the AGN and SF galaxies. The results of a linear regression analysis to quantify any correlation between the spectral index and radio luminosity for both populations are given in Table 7. For the AGN, there is a statistically significant anti-correlation between $\alpha_{\mathrm{low}}$ and $P_{200\,\mathrm{MHz}}$ , and $\alpha_{\mathrm{high}}$ and $P_{200\,\mathrm{MHz}}$ ; $\alpha_{\mathrm{low}}$ and $\alpha_{\mathrm{high}}$ steepen by $0.062 \pm 0.011$ and $0.037 \pm 0.011$ per logarithmic luminosity decade, respectively. For the SF galaxies, there is no statistically significant correlation between $\alpha_{\mathrm{low}}$ and $P_{200\,\mathrm{MHz}}$ , and $\alpha_{\mathrm{high}}$ and $P_{200\,\mathrm{MHz}}$ .

Table 8. Local RLFs at 200 MHz for all radio sources, the AGN and SF galaxies in the GLEAM-6dFGS sample.

Figure 12. Left: radio colour–colour diagram for all AGN in the GLEAM-6dFGS sample with measured $\alpha_{\mathrm{low}}$ and $\alpha_{\mathrm{high}}$ ; $\alpha_{\mathrm{low}}$ is the spectral index between 76 and 227 MHz, and $\alpha_{\mathrm{high}}$ is the spectral index between 200 and 1 400/843 MHz. AGN classed as Aa are shown in red, Ae in blue and Aae in cyan. The dashed lines represent spectral indices of zero and equal values of $\alpha_{\mathrm{low}}$ and $\alpha_{\mathrm{high}}$ . Individual error bars are not plotted for clarity, but the median error bar size for the sample is shown at the bottom right. Right: radio colour–colour diagram for all SF galaxies in the GLEAM-6dFGS sample with measured $\alpha_{\mathrm{low}}$ and $\alpha_{\mathrm{high}}$ .

Laing & Peacock (Reference Laing and Peacock1980) investigated the relation between spectral index and radio luminosity for a complete sample of 165 sources with $S_{178\,\mathrm{MHz}} > 10$ Jy, consisting primarily of FRII sources. For the extended structure in the FRII sources, $\alpha_{0.75\,\mathrm{GHz}}^{5\,\mathrm{GHz}}$ was found to be anti-correlated with the 1.4-GHz radio luminosity. The slope of the regression line (i.e. the change in the spectral index per logarithmic luminosity decade) was measured as –0.088. A similar anti-correlation was found between $\alpha_{0.75\,\mathrm{GHz}}^{5\,\mathrm{GHz}}$ and redshift, which is expected given the strong correlation between luminosity and redshift in bright, flux-limited samples. The $P-\alpha$ anti-correlation was confirmed by Blundell, Rawlings, & Willott (Reference Blundell, Rawlings and Willott1999) for a number of complete samples of FRII sources selected at frequencies close to 151 MHz. They suggested that the anti-correlation is caused by radiative losses in the enhanced magnetic fields of the hotspots of sources with more powerful jets.

We find that the $P-\alpha$ anti-correlation continues to hold for less powerful sources in the local universe. de Gasperin, Intema, & Frail (Reference de Gasperin, Intema and Frail2018) investigated the spectral index properties between 147 and 1 400 MHz of radio sources from TGSS and NVSS, and reported a flattening of the median spectral index with decreasing flux density. A similar spectral flattening towards lower flux densities has been observed in deeper radio samples covering smaller sky areas (e.g. Prandoni et al., Reference Prandoni, Parma, Wieringa, de Ruiter, Gregorini, Mignano, Vettolani and Ekers2006; Intema et al., Reference Intema, van Weeren, Röttgering and Lal2011; Whittam et al., Reference Whittam2013). Since compact radio cores display much flatter spectra and are typically much fainter than the lobes of well-evolved radio galaxies, de Gasperin et al. argued that this trend is driven by the presence of core-dominated and young AGN at low flux densities. Our results for the GLEAM-6dFGS AGN are consistent with this picture. Flat-spectrum sources are more common at lower radio luminosity, suggesting the existence of a significant population of low-luminosity AGN that remain core-dominated even at low frequencies.

Figure 14 shows that there is a strong increase in the fraction of sources with optical emission lines with $\alpha_{\mathrm{low}}$ and $\alpha_{\mathrm{high}}$ . In other words, optical emission lines are much more common in galaxies which host flat-spectrum radio sources. In their study of the local radio source population at 20 GHz, Sadler et al. (Reference Sadler, Ekers, Mahony, Mauch and Murphy2014) observed the same trend for radio galaxies with weak optical emission lines (class Aae) when measuring spectral indices between ${\sim}1$ and 20 GHz. The fraction of radio galaxies with strong emission lines (class Ae) was not found to change significantly with spectral index.

Figure 13. $\alpha_{\mathrm{low}}$ (red) and $\alpha_{\mathrm{high}}$ (blue) as a function of the 200-MHz radio luminosity for the AGN (left) and SF galaxies (right) in the GLEAM-6dFGS sample with measured $\alpha_{\mathrm{low}}$ and $\alpha_{\mathrm{high}}$ . The dashed lines are the linear regression lines.

Figure 14. Top: fractions of AGN and SF galaxies as a function of $\alpha_{\mathrm{low}}$ (left) and $\alpha_{\mathrm{high}}$ (right). Bottom: among the AGN, fractions of Aa, Aae, and Ae sources as a function of $\alpha_{\mathrm{low}}$ (left) and $\alpha_{\mathrm{high}}$ (right).

Zajaček et al. (Reference ZajaČek2019) studied radio spectral index trends of SDSS-FIRST sources across optical emission line diagnostic diagrams to search for potential correlations. They found a trend of decreasing 4.85–10.45 GHz spectral index with increasing ionisation ratio [OIII]/H $\beta$ , which corresponds to the LINER-Seyfert transition in the optical diagnostic diagrams. They interpreted this trend as a result of re-triggered nuclear and jet activity.

In low-frequency selected samples, there is preferential detection of sources with ultra-steep spectra down to low frequencies. A significant fraction of AGN in the GLEAM-6dFGS sample have ultra-steep spectra: 4.2% have $\alpha_{\mathrm{low}} < -1.2$ and 4.6% have $\alpha_{\mathrm{high}} < -1.2$ . Mahatma et al. (Reference Mahatma2018) reported a similar fraction (4.1%) of sources with $\alpha_{150}^{1\,400} < -1.2$ in their LOFAR study of remnant radio-loud AGN in the Herschel-ATLAS field. The ultra-steep-spectrum AGN in our sample have a wide range of radio luminosities; the median value of $P_{200\,\mathrm{MHz}}$ for the AGN with $\alpha_{\mathrm{low}} < -1.2$ ( $10^{24.41}\,\mathrm{W}\,\mathrm{Hz}^{-1}$ ) is slightly lower than the median value of $P_{200\,\mathrm{MHz}}$ for all AGN ( $10^{24.61}\,\mathrm{W}\,\mathrm{Hz}^{-1}$ ). Quici et al. (Reference Quici2021) show that remnant radio galaxies can be reliably identified by their ultra-steep spectra at low frequency ( ${\lesssim}400$ MHz), although the technique fails to capture younger remnants with steeper spectra at high frequencies. The absence of a radio core at high frequency is needed to establish whether these are genuine remnant radio galaxies.

5.1. Calculating the local RLF

We calculate the local RLF using the $1/V_{\mathrm{max}}$ method of Schmidt (Reference Schmidt1968), where $V_{\mathrm{max}}$ is the maximum volume within which a galaxy can satisfy all the sample selection criteria ( $S_{200\,\mathrm{MHz}} > 100$ mJy in the shallow region, $S_{200\,\mathrm{MHz}} > 55$ mJy in the deep region, and $K \leq 12.65$ mag in both the shallow and deep regions). The luminosity function in a given luminosity bin centred at L is given by:

(10) \begin{equation}\Phi(L) = \sum\limits_{i=1}^N \left( \frac{1}{V_{\mathrm{max,\,deep},\,i} + V_{\mathrm{max,\,shallow},\,i} } \right) \,,\end{equation}

where N is the number of galaxies in the luminosity bin, and $V_{\mathrm{max,\,deep},\,i}$ and $V_{\mathrm{max,\,shallow},\,i}$ are the maximum volumes within which the $i^{\mathrm{th}}$ galaxy can satisfy the sample selection criteria in the deep and shallow regions respectively. The rms Poisson counting error in $\Phi(L)$ is given by:

(11) \begin{equation}d\Phi(L) = \left[ \sum\limits_{i=1}^N \left( \frac{1}{V_{\mathrm{max,\,deep},\,i} + V_{\mathrm{max,\,shallow},\,i} } \right)^2 \right]^{1/2} \,.\end{equation}

We calculate the luminosity function in 13 luminosity bins of width 1 mag ( $=0.4\,\mathrm{dex}$ ) between $10^{21.6}$ and $10^{26.8}\,\mathrm{W}\,\mathrm{Hz}^{-1}$ , and one luminosity bin of width 2 mag between $10^{26.8}$ and $10^{27.6}\,\mathrm{W}\,\mathrm{Hz}^{-1}$ . In luminosity bins where the number, N, of galaxies is small ( $N < 5$ ), we use 84% confidence upper and lower limits based on Poisson statistics, tabulated in Gehrels (Reference Gehrels1986).

Following Condon et al. (Reference Condon, Matthews and Broderick2019), we correct the measured luminosity function for galaxy clustering at a distance s centred on our own Galaxy using

(12) \begin{equation}\frac{\Phi_{\mathrm{P}}}{\Phi} = 1 + \frac{3}{3 - \gamma_{\mathrm{s}}} \left( \frac{s_{0}}{s} \right)^{\gamma_{\mathrm{s}}} \,,\end{equation}

where $\frac{\Phi_{\mathrm{P}}}{\Phi}$ is the expected overdensity near our own Galaxy, or the space density, $\Phi_{\mathrm{P}}$ , of local galaxies divided by the average space density, $\Phi$ , of all galaxies, $\gamma_{\mathrm{s}} = 1.66$ and $s_{0} = 3.76 h^{-1}$ Mpc (Fisher et al., Reference Fisher, Davis, Strauss, Yahil and Huchra1994). The volume within s is multiplied by $\frac{\Phi_{\mathrm{P}}}{\Phi}$ when calculating $V_{\mathrm{max}}$ ; the correction is only applied for $s < 20 h^{-1}$ Mpc.

The completeness of the GLEAM-6dFGS cross-matched sample is estimated to be 95% (see Section 3.1.1). To correct the luminosity function for incompleteness, we therefore multiply $\Phi(L)$ in all bins by $\frac{1}{0.95}$ . Additionally, we need to account for spectroscopic incompleteness in the 6dFGS catalogue, that is, the fact that some galaxies in the 6dFGS target list were either not able to be observed or had only a poor-quality spectrum so that a reliable redshift could not be measured. The 6dFGS spectroscopic completeness varies across the sky but is estimated to be 85% for the NVSS/SUMSS-6dFGS sample as a whole. To correct the luminosity function for spectroscopic incompleteness, we therefore multiply $\Phi(L)$ in all bins by a further factor of $\frac{1}{0.85}$ .

5.3. Comparison of recent RLF measurements

In the left panel of Figure 16, we compare the local RLF for AGN measured in this paper with the local RLF at 150 MHz for AGN from the LoTSS-SDSS sample (Sabater et al., Reference Sabater2019), extrapolated to 200 MHz assuming $\alpha = -0.7$ , the typical spectral index at low frequency of AGN measured in this paper. Overall, the luminosity functions are in good agreement. The median redshift (0.143) of the AGN in the LoTSS-SDSS sample is higher than that (0.081) in the GLEAM-6dFGS sample. Cosmological evolution of HERGs, which dominate the local AGN population at $P_{1.4\,\mathrm{GHz}} \gtrsim 10^{26}\,\mathrm{W}\,\mathrm{Hz}^{-1}$ (see e.g. Heckman & Best, Reference Heckman and Best2014; Pracy et al., Reference Pracy2016), may therefore partly explain the slight offset between the luminosity functions at the high-luminosity end.

We use the parameterised form of the 1 400-MHz local RLF for AGN from equation (28) of Condon et al. (Reference Condon, Matthews and Broderick2019) and fit this to the 200-MHz data by making a shift in radio power set by a single characteristic 200–1 400 MHz spectral index. The best-fitting value of $\alpha$ is –0.40, which is flatter than the median value of $\alpha_{\mathrm{high}}$ (–0.60) for the AGN in the GLEAM-6dFGS sample. The AGN sample by Condon et al. (Reference Condon, Matthews and Broderick2019) has a slightly higher median redshift (0.12). Any luminosity evolution of the AGN population would therefore cause the best-fitting value of $\alpha$ to be flatter than expected. We note, however, that the local AGN population is dominated by LERGs at $P_{1.4\,\mathrm{GHz}} \lesssim 10^{26}\,\mathrm{W}\,\mathrm{Hz}^{-1}$ , which display little or no evolution (see e.g. Heckman & Best, Reference Heckman and Best2014; Pracy et al., Reference Pracy2016). There is a significant change in the shape of the luminosity function. At lower luminosity, a flatter spectral index is needed to obtain a good fit. This is consistent with our earlier investigation of spectral index versus luminosity in Section 4.3, where we found that the typical value of $\alpha_{\mathrm{high}}$ flattens with decreasing luminosity.

In Figure 17, we measure the local RLF for the AGN in the GLEAM-6dFGS sample, separating between flat-spectrum ( $\alpha_{\mathrm{high}} > -0.5$ ) and steep-spectrum ( $\alpha_{\mathrm{high}} < -0.5$ ) sources. While steep-spectrum sources overwhelmingly dominate the luminosity function at the highest luminosities, the relative contribution to the luminosity function from flat-spectrum sources gradually increases with decreasing luminosity.

Figure 16. Left: the local RLF at 200 MHz for AGN measured in this paper (black circles). The local RLF at 150 MHz for AGN from Sabater et al. (Reference Sabater2019) extrapolated to 200 MHz assuming a spectral index of –0.7 (red circles). The local RLF at 1 400 MHz for AGN using the parameterisation of Condon et al. (Reference Condon, Matthews and Broderick2019), extrapolated to 200 MHz assuming spectral indices of 0 (solid blue line), –0.40 (dashed blue line), and –0.7 (dot-dashed blue line), as discussed in the text. Right: the local RLF at 200 MHz for SF galaxies measured in this paper (black circles). The local RLF at 150 MHz for SF galaxies from Sabater et al. extrapolated to 200 MHz assuming a spectral index of –0.6 (red circles). The local RLF at 1 400 MHz for SF galaxies using the parameterisation of Condon et al. (Reference Condon, Matthews and Broderick2019), extrapolated to 200 MHz assuming spectral indices of 0 (solid blue line), –0.66 (dashed blue line), and –0.8 (dot-dashed blue line), as discussed in the text.

Figure 17. The local RLF for the GLEAM-6dFGS AGN, separating between flat-spectrum (blue points) and steep-spectrum (red points) sources. The dashed curve shows the double power-law fit to the local RLF for all AGN. There is an increase in the relative contribution of flat-spectrum sources with decreasing luminosity.

In the right panel of Figure 16, we compare the local RLF for SF galaxies measured in this paper with the local RLF at 150 MHz for SF galaxies by Sabater et al. (Reference Sabater2019), extrapolated to 200 MHz assuming $\alpha = -0.6$ , the typical spectral index at low frequency of SF galaxies measured in this paper. The luminosity function by Sabater et al. is shifted by $\log_{10} P_{200\,{\textrm{MHz}}} \approx 0.1$ to the right relative to our measurements. As mentioned in Section 1, the LoTSS-SDSS sample covers a much smaller area of sky ( $424\,\mathrm{deg}^{2}$ ) and extends to much lower flux densities ( ${\approx}0.4$ mJy) than the GLEAM-6dFGS sample. Consequently, the median redshift (0.097) of the SF galaxies in the LoTSS-SDSS sample is significantly higher than that (0.015) in the GLEAM-6dFGS sample. Ocran et al. (Reference Ocran, Taylor, Vaccari, Ishwara-Chandra, Prandoni, Prescott and Mancuso2020) studied the properties of SF galaxies selected at 610 MHz with the GMRT in a deep survey covering ${\approx}1.86\,\mathrm{deg}^{2}$ . They explored the evolution of the RLF for SF galaxies out to $z \approx 1.5$ . The SF galaxies were found to evolve as $f(z) = (1+z)^{(2.95 \pm 0.19)-(0.50 \pm 0.15)z}$ assuming pure luminosity evolution. The luminosity function by Sabater et al. is therefore expected to be shifted to the right by $\log_{10} P_{200\,{\textrm{MHz}}} \approx \log_{10} [ f(0.097)/f(0.015) ] = 0.10$ relative to our measurements. We conclude that the luminosity functions are in good agreement after taking the rapid evolution of the SF galaxy population into account.

We use the parameterised form of the 1 400-MHz local RLF for SF galaxies from equation (28) of Condon et al. (Reference Condon, Matthews and Broderick2019) and fit this to the 200-MHz data using a single characteristic 200–1 400 MHz spectral index. The median redshift of the SF galaxies in the NVSS-2MASX sample by Condon et al. is 0.06. The best-fitting value of $\alpha$ is –0.66, which is very close to the median value of $\alpha_{\mathrm{high}}$ (–0.65) for the SF galaxies in our sample.