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Appendix A: Notes on peculiar 3 GHz sources

AzTEC2, 5, and 8 each show, in addition to a 3 GHz source coinciding with the submm peak, an additional source lying at SW, NE, and NE from the SMA peak, respectively. The additional 3 GHz feature towards AzTEC2 could, in principle, represent a radio-emitting lobe of a jet interacting with the surrounding medium or a merger component (projected separation is 12.7 proper kpc at the redshift of AzTEC2). We note that the 1.3 mm emission detected towards AzTEC2 with ALMA also shows an additional weak (~2.9σ) feature at to the SW of AzTEC2 (Cycle 2 ALMA project 2013.1.00118.S; M. Aravena et al., in prep.), but its peak position lies away to the SW of the 3 GHz feature; this offset is within the large statistical positional uncertainty of ~1″ of the weak 1.3 mm feature (Δθ_stat ∝ (S/N)^-1).

The additional radio source towards AzTEC5 was already seen at 1.4 GHz (see Table 4), and it is also visible in the Spitzer/IRAC and MIPS images (Younger et al. 2007, Fig. 1 therein). This source can be associated with the Herschel-detected emission-line galaxy 150.08336+02.53619, for which a spectroscopic redshift of z_spec(Hα) = 1.42 was reported by Roseboom et al. (2012). Also, a 5.4σ detection with ALMA at 994 μm is obtained towards this source (Cycle 1 ALMA project 2012.1.00978.S; PI: A. Karim; O. Miettinen et al., in prep.), while the ALMA 1.3 mm detection is of ~3σ significance (M. Aravena et al., in prep.). We note that the most up-to-date COSMOS spec-z catalogue gives a lower redshift value of z_spec = 0.9044, which is based on observations with the Inamori Magellan Areal Camera and Spectrograph (IMACS; Salvato et al., in prep.); however, the quality flag is 1, i.e. this z_spec is considered insecure. Hence, the 3 GHz source NE of AzTEC5 is probably a lower-redshift galaxy (AzTEC5 itself has ; see Table 1). Similarly, the two radio sources towards AzTEC8 were already seen at 1.4 GHz (Younger et al. 2009, Fig. 1 therein; see our Table 4): the 1.4 GHz source to the NE of the SMA-detected SMG was called AzTEC8.E by Younger et al. (2009), and the Spitzer 24 μm emission towards AzTEC8 is coincident with AzTEC8.E. This source can be associated with the Chandra/X-ray -detected galaxy CXOC J095959.5+023441 whose photo-z is z_phot = 2.420 ± 0.060 (Salvato et al. 2011), hence it is probably at a lower redshift than the SMG on its western side (z_spec = 3.179). This source is also detected with the VLBA at 1.4 GHz at milliarcsec resolution (S_{1.4 GHz} = 83.8 μJy), showing the presence of a radio-emitting AGN (Herrera Ruiz et al., in prep.; Herrera Ruiz, priv. comm.).

Towards AzTEC11, we have detected a double 3 GHz source, projectively separated by , and where the southern component is coincident with AzTEC11-N⁷. The northern 3 GHz source is nearly equidistant from AzTEC11-N ( separation) and AzTEC11-S ( separation). Younger et al. (2009) reported that the calibrated visibility data of AzTEC11 show significant structure and are best modelled with a double point source (their Table 2). The complex structure of the visibility data probably makes the derived source positions to be rather uncertain. However, a positional uncertainty of only in both right ascension and declination for the 890 μm peak of AzTEC11-N and 11-S was reported by Younger et al. (2009; Table 1 therein). The authors also recognised an elongated 1.4 GHz source towards AzTEC11, where the emission morphology resembles that seen at 890 μm. Their two-component Gaussian fit yielded comparable 1.4 GHz flux densities for the two sources (see our Table 4). The two 3 GHz sources seen towards AzTEC11 share a common radio envelope (at the 3σ level), and are probably in the process of merging⁸. The ALMA 1.3 mm observations at resolution towards AzTEC11 revealed two SMGs separated by in projection (M. Aravena et al., in prep.), and the northern one is well coincident with our northern 3 GHz source (only offset).

Towards AzTEC15 the projected separation between the 890 μm and 3 GHz positions is fairly large, i.e. . The 890 μm detection of AzTEC15 by Younger et al. (2009) was only of moderate significance (4.4σ), and no 1.4 GHz counterpart was detected, but it was found to be associated with Spitzer IR emission. The 890 μm position uncertainty reported by Younger et al. (2009) is in right ascension and in declination. The resolution ALMA 1.3 mm observations towards AzTEC15 (M. Aravena et al., in prep.), however, show a perfect positional coincidence ( offset) with our 3 GHz source of 5.4σ significance, and hence it is physically related to AzTEC15.

The 3 GHz source near AzTEC21a ( NE of the PdBI position) is our weakest source with a S/N of 4.2. The reliability of this 3 GHz source candidate is supported by the fact that it is also seen at 1.4 GHz, although the 1.4 GHz source is also weak (peak surface brightness of 63 μJy beam^-1 or ~3.9σ; see Miettinen et al. 2015, Fig. A.1 therein). Hence, we include the 3 GHz source near AzTEC21a in our radio size analysis.

The 3 GHz source (12.6σ) detected SW in projection from AzTEC24b is also detected at 1.4 GHz (Table 4), and the 1.4 GHz source was associated with the ASTE/AzTEC 1.1 mm SMG AzTEC/C48 by Aretxaga et al. (2011), a source also detected with Herschel (see Miettinen et al. 2015). There is also a 1.3 mm-detected ALMA source lying SW from our PdBI detection and SE from the 3 GHz source in question (Aravena et al., in prep.). The ALMA detection in particular shows that the 3 GHz source is an SMG despite the relatively large separation from our PdBI source. We note that there is a 4.3σ 3 GHz source lying N of AzTEC27. Miettinen et al. (2015) reported the presence of weak 1.4 GHz emission (peak intensity of 32.1 μJy beam^-1 or ~2.5σ) towards this position, but the nature of this radio emission remains unclear (i.e. noise feature or associated with the SMG). The weak 3 GHz source associated with AzTEC27 is included in the present radio size analysis. The additional 3 GHz radio sources not analysed further in the present study are described in Appendix B, and those not detected at 3 GHz are discussed in Appendix C.

Appendix B: Notes on additional 3 GHz sources

As shown in Fig. 2, SW from AzTEC20, NW from AzTEC22, and E from AzTEC23, there is a clearly detected 3 GHz source (10.3σ, 22.8σ, and 5.5σ, respectively). Interestingly, these 3 GHz sources are closer to the original AzTEC 1.1 mm positions than to the PdBI source candidates ( E, SW, and NE from AzTEC20, 22, and 23; see Scott et al. 2008; Miettinen et al. 2015). Morever, the 3 GHz sources detected towards the AzTEC20, 22, and 23 fields have a Herschel 250 μm detection lying at SW, SE, and NW, respectively (as based on the cross-correlation with the COSMOS SPIRE 250 μm Photometry Catalogue). These radio sources are not analysed further in the present study.

Appendix C: Notes on the 3 GHz non-detections’ appearances at other wavelengths

The following 21 SMGs (54 ± 12% of the whole sample) were not detected at 3 GHz: AzTEC10, 13, 14-E, 14-W, 16, 17b, 18, 19b, 20, 21b, 21c, 22, 23, 24a, 24c, 26a, 26b, 28, 29a, 29b, and 30.

AzTEC10, 13, 14-E, and 14-W were detected at 890 μm with a S/N of 5.3, 4.6, 5.0, and 3.9, respectively, by Younger et al. (2009). Only the most significant of these moderate SMA 890 μm detections, namely AzTEC10, was found to exhibit Spitzer IR emission, while none of them was detected at optical wavelengths or at 1.4 GHz (Younger et al. 2009). Hence, their non-detection at 3 GHz is not surprising.

As discussed by Miettinen et al. (2015; Appendix C therein), the PdBI 1.3 mm SMG candidates AzTEC16, 17b, 20, 21c, 22, 24a, 24c, 26b, 29a, and 30 have no multiwavelength counterparts and some of them could be spurious. The 1.3 mm S/N of these sources were found to be in the range of 4.5–6. Moreover, AzTEC29b, although a 7.3σ detection, was found to lie at edge of the 1.3 mm map. On the other hand, AzTEC27 and AzTEC28 (S/N_{1.3 mm} = 6 and 5.5, respectively) were among the best PdBI detections by Miettinen et al. (2015), but neither of them were found to have multiwavelength counterparts; only a trace of 1.4 GHz emission (2.5σ) was seen towards AzTEC27 (Appendix A). AzTEC18, 19b, 21b, 21c, 23, and 26a were detected with S/N_{1.3 mm} = 4.2–9.7, and only AzTEC21c was found to not have multiwavelength counterparts. However, none of these SMGs was detected at 1.4 GHz. Given the aforementioned properties, it comes as no surprise that among AzTEC16–30 there are so many 3 GHz non-detections (17 of 22, i.e. 77 ± 19%).

Appendix D: Testing the reliability of the size measurements

To test the reliability of our FWHM size measurements, we simulated sources using the CASA (release 4.3.1) Toolkit. We first generated mock galaxies with a Gaussian flux distribution, intrinsic FWHM size fixed to (i.e. the average deconvolved FWHM derived for our SMGs where both the major and minor axes could be determined or constrained), PA ranging from 0° to 135° in steps of 45°, and flux densities corresponding to S/N in the range of 4 to 38 in steps of S/N = 2, which cover the observed range of S/N of our SMGs (S/N = 4.2–37.4). The simulated image was convolved to the resolution of to match the resolution of our real data. To obtain a realistic background noise level, the simulated galaxies were added to a 3 GHz map of in size, and which was cropped from a source-free region of COSMOS (1σ = 2.3 μJy beam^-1). The resulting image is shown in the top panel in Fig. D.1. The deconvolved FWHM sizes of the simulated sources were then determined using the AIPS task JMFIT as described in Sect. 4.1.

The bottom panel in Fig. D.1 shows the ratio of the measured size to the input size as a function of the S/N. The data points are shown separately for the major and minor axes. As expected, the size measurement is generally more accurate for more significant detections, but within the size uncertainties determined by JMFIT the measured deconvolved sizes are in good agreement with the real intrinsic sizes (see the dashed line in Fig. D.1 indicating the one-to-one correspondence). Because most of our detections are of high significance (median S/N = 12.6), these simulations suggest that our size measurements are reliable.

Fig. D.1 Top: simulated SMGs added to a noise map to simulate our real 3 GHz VLA data. The S/N of the sources increases from left to right, top to bottom (S/N = 4–38), and in each row the PA ranges from 0° to 135° in steps of 45° (being 0° and 45° for the two bottommost objects). The synthesised beam size of is shown in the bottom left corner. Bottom: ratio of the measured to the simulated input source size (deconvolved FWHM) as a function of the S/N. The green points show the major axes ratio, while the red points show that between the minor axes. The minor axis FWHM for the faintest source (S/N = 4) could not be determined by the AIPS task JMFIT. The vertical error bars were propagated from the size errors determined by JMFIT. A one-sided error bar is shown for those cases where the minimum size could not be determined by JMFIT. The horizontal dashed line shows the line of equality between the sizes.

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Appendix E: Calculation of the cosmic-ray electron cooling times

To quantitatively examine the possibility that the observed radio-continuum sizes of SMGs could be the result of CR diffusion, we first calculate the maximum lifetime of the electrons considering the radiative energy losses due to synchrotron emission, inverse Compton (IC) scattering, bremsstrahlung, and ionisation processes. In what follows, we calculate the corresponding cooling times using the formulas from Murphy (2009; Sect. 2 therein) to which we refer the reader for a more detailed description.

The redshifts of our 3 GHz detected SMGs range from z_spec = 0.834 for AzTEC17a to z_spec = 5.298 for AzTEC3. If we assume that the critical frequency at which the electron emits most of its energy, ν_crit, is equal to ν_rest = ν_obs(1 + z), its value is in the range of 5.5 GHz to 18.9 GHz. If the electrons are spiralling in a magnetic field whose strength is B ~ 100 μG (a starburst-type B-field; e.g. Lacki & Beck 2013), the relationship ν_crit = 1.3 × 10^-2(B/μG)(E/ GeV)² yields CR electron energies of E ≃ 2–3.8 GeV. For this case, the synchrotron cooling timescale for CR electrons is τ_syn ~ 1.4 × 10⁹(ν_crit/ GHz)^{− 1 / 2}(B/μG)^{− 3 / 2} ~ 3.2–6 × 10⁵ yr.

The IC cooling timescale is given by τ_IC ~ 5.7 × 10⁷(ν_crit/ GHz)^{− 1 / 2}(B/μG)^{1 / 2}(U_rad/ 10^-12 erg cm^-3)^-1 yr, where U_rad is the radiation field energy density of the galaxy. To estimate the value of U_rad, we adopt a total infrared (8–1 000 μm) luminosity range of L_IR ~ 10¹²−10¹³L_⊙ appropriate for SMGs (e.g. Magnelli et al. 2012; Swinbank et al. 2014; da Cunha et al. 2015; see Miettinen et al., in prep., for the present SMG sample), and as the characteristic size we use the median 3 GHz major axis FWHM size derived here, i.e. 4.2 kpc, which corresponds to a circular area of 13.9 kpc². Using Eq. (4) of Murphy et al. (2012a), these values imply U_rad in the range of ~5 × 10^-10–4.9 × 10^-9 erg cm^-3; for a smaller IR-emitting area (A_IR), the value of would be higher. The values of ν_crit and B being as above, we derive τ_IC ~ 2.6 × 10⁴–4.9 × 10⁵ yr. In the context of IC cooling, it should be noted that our SMG sample contains high-redshift sources, the most extreme case being AzTEC3 at z_spec = 5.298. At high redshifts, the IC scattering between relativistic electrons and the cosmic microwave background (CMB) – boosting the CMB photon energy – becomes more important compared to the low-z universe. The reason for this is that the energy density of the CMB increases steeply with redshift, namely U_CMB ∝ (1 + z)⁴. For instance, at z_spec = 5.298, the CMB photon bath has about 140 times higher energy density than that at the lowest-redshift SMG in our sample (AzTEC17a at z_spec = 0.834). This means that besides the intense IR radiation field in a starburst, IC losses off the CMB photons have the potential to increase the cooling rate of the CR electrons (e.g. Lacki & Thompson 2010).

The bremsstrahlung lifetime is τ_brem ~ 8.6 × 10⁷(n_H/cm^-3)^-1 yr, where n_H is the hydrogen number density of the ISM. Assuming a typical average density range of n_H = 10²–10³ cm^-3, we obtain τ_brem ~ 8.6 × 10⁴–8.6 × 10⁵ yr⁹.

The timescale for ionisation losses can be written as τ_ion ~ 3.6 × 10¹⁰(ν_crit/ GHz)^{1 / 2}(B/μG)^{− 1 / 2}(n_H/ cm^-3)^-1 × [3 / 2 × ln(ν_crit/B) + 49]^-1 yr, which under our assumptions lies in the range of τ_ion ~ 1.9 × 10⁵–3.4 × 10⁶ yr.

Finally, due to the combined energy losses from the aforementioned processes, the effective cooling lifetime for CR electrons is given by (E.1)The individual timescales calculated above yield τ_cool ~ 1.7 × 10⁴–1.9 × 10⁵ yr. In the case of random-walk diffusion, the electrons’ escape scale-length is given by l_esc = (D_Eτ_esc)^{1 / 2}, and when the diffusion coefficient D_E is in the energy-dependent regime (E ≥ 1 GeV), the escape length is given by l_esc ~ 7.1 × 10^-4(τ_esc/ yr)^{1 / 2}(ν_crit/ GHz)^{1 / 2}(B/μG)^{− 1 / 2} kpc. During the above derived cooling time (τ_esc = τ_cool) the electrons can travel only l_esc ~ 22–135 pc. Hence, we conclude that if the FIR/star-forming sizes of our SMGs are as compact as those from Simpson et al. (2015a; 2.4 ± 0.2 kpc in median FWHM) and Ikarashi et al. (2015; ~1.6 kpc in median FWHM), it seems unlikely that CR electrons would have had time to propagate from their sites of origin to the large distances where we observe the 3 GHz emission [the median major axis FWHM size being 4.2 ± 0.9 kpc]. In the above analysis we did not add the effect of the IC scattering off the CMB radiation, although at high redshift it can shorten the electron lifetime and diffusion length scale even more. However, apart from AzTEC3, our Fig. 7 does not show evidence of smaller radio sizes at higher redshifts as expected if the electrons have less time to travel away from their site of origin. This is possibly a manifestation of the fact that in starburst galaxies, at whatever redshift they might be, the local stellar radiation field is intense (cf. the above estimate), and hence the CR electrons can suffer from strong IC losses from stellar light besides/instead from the CMB (e.g. Lisenfeld et al. 1996; Lacki & Thompson 2010). Moreover, we have ignored the fact that if the galaxy is associated with a galactic-scale wind, the CR particles in the wind adiabatically lose momentum and energy on the course of the expansion of the wind (e.g. Völk et al. 1996). The effect of losses due to electron advection would further shorten the diffusion length-scale.

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