THE DISCOVERY OF LENSED RADIO AND X-RAY SOURCES BEHIND THE FRONTIER FIELDS CLUSTER MACS J0717.5+3745 WITH THE JVLA AND CHANDRA

JVLA observations of MACS J07175+3745 were obtained in the L-band (1–2 GHz) in the A-array configuration, in the S-band (2–4 GHz) in the A- and B-array configurations, and in the C-band (4.5–6.5 GHz) in the B-array configuration. All observations were done using single 6 hr runs, resulting in a typical on-source time of ∼5 hr. The total recorded bandwidth was 1 GHz for the L-band, and 2 GHz for the S- and C-bands. The primary calibrators were 3C138 and 3C147. The secondary calibrator (J0713+4349) was observed for a couple of minutes at 30–40 minute intervals. All four circular polarization products were recorded. An overview of the observations is given in Table 1.

The data were reduced with CASA version 4.2.1 (McMullin et al. 2007). The data from the different observing runs were all reduced in the same way. For the two primary calibrators, we used the clean-component models provided by CASA. We also took the overall spectral index of the primary calibrator sources into account, scaling the flux density for each channel.

As a first step, the data were Hanning smoothed and pre-determined elevation-dependent gain tables and antenna position offsets were applied. This was followed by automatic flagging of radio frequency interference (RFI) using the tfcrop mode of the CASA task flagdata. We then determined an initial bandpass correction using 3C147. This bandpass was applied and additional RFI was identified and flagged with the AOFlagger (Offringa et al. 2010). The reason for applying the bandpass is to avoid flagging good data due to the bandpass roll off at the edges of the spectral windows.

Next, we determined complex gain solutions on 3C147 for the central 10 channels of each spectral window to remove possible time variations of the gains during the calibrator observations. We pre-applied these solutions to find the delay terms (gaintype = "K") and bandpass calibration. Applying the bandpass and delay solutions, we re-determined the complex gain solutions for both primary calibrators using the full bandwidth. We then determined the global cross-hand delay solutions (gaintype = "KCROSS") from the polarized calibrator 3C138. For 3C138, we assumed a rotation measure (RM) of 0 rad m⁻¹, and for the RL-phase difference we took −15°. All relevant solutions tables were applied on the fly to determine the complex gain solution for the secondary calibrator J0713+4349 and to establish its flux density scale accordingly. The absolute flux scale uncertainty due to bootstrapping from the calibrators is assumed to be a few percent (Perley & Butler 2013). As a next step, we used J0713+4349 to find the channel-dependent polarization leakage terms. 3C138 was used to perform the polarization angle calibration for each channel.²² Finally, all solutions were applied to the target field. The corrected data were then averaged by a factor of 3 in time and a factor of 6 in frequency.

To refine the calibration for the target field, we performed three rounds of phase-only self-calibration and a final round of amplitude and phase self-calibration. For the imaging we employed w-projection (Cornwell et al. 2008, 2005) to take the non-coplanar nature of the array into account. Image sizes of up to 12288² pixels were needed (A-array configuration) to deconvolve a few bright sources outside the main lobe of the primary beam. For each frequency band, the full bandwidth was used to make a single deep Stokes I continuum image. We used Briggs (1995) weighting with a robust factor of 0. The spectral index was taken into account during the deconvolution of each observing run (nterms = 3; Rau & Cornwell 2011). We manually flagged some additional data during the self-calibration process by visually inspecting the self-calibration solutions. Clean masks were employed during the deconvolution. The clean masks were made with the PyBDSM source detection package (Mohan & Rafferty 2015). The S-band A-array and B-array configuration data were combined after the self-calibration to make a single deep 2–4 GHz image. The final images were made with Briggs weighting and a robust factor of 0.75, except for the C-band image, for which we employed natural weighting. Images were corrected for the primary beam attenuation, with the frequency dependence of the beam taken into account. An overview of the resulting image properties, such as rms noise and resolution, is given in Table 2.

MACS J0717.5+3745 was observed with Chandra for a total of 243 ks between 2001 and 2013. A summary of the observations is presented in Table 3. The data sets were reduced with CIAO v4.7 and CALDB v4.6.5, following the same methodology that was described by Ogrean et al. (2015).

Point sources were detected with the CIAO script wavdetect in the energy bands 0.5–2 and 2–7 keV, using wavelet scales of 1, 2, 4, 8, 16, and 32 pixels and ellipses with radii $5\sigma$ around the centers of the detected sources. Due to the complicated morphology of MACS J0717.5+3745, all sources found by wavdetect were visually inspected and some false detections associated with the extended ICM of the cluster were removed. The local background around each point source was described using an elliptical annulus with an inner radius equal to the source radius and an outer radius approximately ∼3 times larger than the source radius. To model the X-ray spectra of the point sources, the local background spectra were subtracted from the corresponding source spectra. The spectra were binned to a minimum of one count/bin, and modeled with Xspec v12.8.2 using the extended C-statistics²³ (Cash 1979; Wachter et al. 1979). All the source spectra were modeled as power laws and included galactic absorption. The hydrogen column density in the direction of MACS J0717.5+3745 was fixed to $8.4\times {10}^{20}$ cm⁻², which is the sum of the weighted average atomic hydrogen column density from the Leiden–Argentine–Bonn (LAB; Kalberla et al. 2005) Survey and the molecular hydrogen column density determined by Willingale et al. (2013) from Swift data.

X-ray fluxes and luminosities were calculated in the energy band 2–10 keV, with uncertainties quoted at the 90% confidence level.

We used the PyBDSM ²⁴ source detection package to find and determine the integrated flux densities of the radio sources in the images. PyBDSM identifies "islands" of contiguous pixels above a detection threshold and fits each island with Gaussians.

For detecting these islands, we took a threshold of $3{\sigma }_{{\rm{rms}}}$ and a pixel threshold of $4{\sigma }_{{\rm{rms}}}$, meaning that at least one pixel in each island needs to exceed the local background by $4{\sigma }_{{\rm{rms}}}$. We determined the local rms noise using a sliding box with a size of 300 pixels to take the noise increase by the primary beam attenuation into account. We manually inspected the output source catalogs and removed any source associated with the radio relic and foreground FR-I source since these sources are larger than the 300 pixel box size for the local noise determination. The source detection was run on all three frequency maps. The locations of the detected sources are indicated on Figure 1.

We then searched for optical counterparts to the radio sources within the area covered by the HST CLASH catalog and HST Frontier Fields observations. For the detected radio sources, we overlaid the radio contours on an HST Frontier Fields color (v1.0²⁵ , F425W, F606W, and F814W band) image to verify that the correct counterparts were identified; see Figures 2 (lensed sources) and 5. Counterparts were found for all radio sources. In a few cases, more than one optical counterpart was identified for the radio source in the CLASH catalog because of the complex morphology of the galaxy in the HST images. An overview of all the compact lensed radio sources that were found within the HST FOV is given in Table 4. The properties of the sources that are not lensed, or for which we could not determine if they were located behind the cluster, are listed in Table 5.

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Table 4.  Properties of Cluster Background Sources

Name/ID	R.A.	Decl.	${S}_{1.5}$	${S}_{3.0}$	${S}_{5.5}$	${z}_{{\rm{phot}}}$	tb	Amplification	X-ray Flux	${z}_{{\rm{spec}}}$
	(°) (J2000)	(°) (J2000)	(μJy)	(μJy)	(μJy)				(10−15 erg cm−2 s−1)	Ebeling et al. (2014)
$S03358$	109.3967560	37.7615846	48.0 ± 11.1	25.9 ± 3.3	11.7 ± 3.9	${1.59}_{-0.06}^{+0.09}$	5.90	3.6 ± 1.0	${17.1}_{-0.20}^{+0.16}$	1.6852 ± 0.001 (E)
S2 2759	109.4097441	37.7666944	⋯	21.2 ± 3.2	16.4 ± 3.8	${0.74}_{-0.03}^{+0.07}$	5.10	1.25 ± 0.8	⋯	⋯
S3 3866	109.4184262	37.7576298	24.4 ± 9.2a	19.4 ± 3.1	11.7 ± 3.6	${1.52}_{-0.07}^{+0.10}$	9.40	1.8 ± 0.4	⋯	⋯
$S64614$	109.4189706	37.7517924	⋯	10.1±1.9b,c	⋯	${1.89}_{-0.09}^{+0.08}$	6.40	2.3 ± 0.7	${8.48}_{-1.08}^{+1.24}$	⋯
$S85637$	109.3935136	37.7423563	⋯	24.0 ± 5.5a	⋯	${1.15}_{-0.03}^{+0.04}$	7.70	6.4 ± 1.7	⋯	⋯
$S96554$	109.3882566	37.7338089	43.7 ± 7.9	20.0 ± 3.0	24.9 ± 3.5	${1.69}_{-0.06}^{+0.22}$	5.60	3.4 ± 1.0	⋯	⋯
S12 7885	109.3850184	37.7221324	96.6 ± 10.5	54.5 ± 3.3	36.9 ± 3.8	${2.32}_{-0.05}^{+0.14}$	5.10	1.8 ± 0.5	⋯	⋯
S21 7107	109.3575070	37.7291308	86.4 ± 8.8	75.7 ± 3.0	87.1 ± 4.6	${1.02}_{-0.06}^{+0.15}$	4.70	1.4 ± 0.3	⋯	⋯
S241 5304	109.3430929	37.7459519	⋯	19.1 ± 3.0	⋯	${0.9}_{-0.4}^{+0.3}$	7.90	1.4 ± 0.2d,e	⋯	⋯
S242 5307	109.3432347	37.7456667	⋯	19.1 ± 3.0	⋯	${0.7}_{-0.3}^{+0.3}$	8.80	1.4 ± 0.2d,e	⋯	⋯
S243 5305	109.3431472	37.7458181	⋯	19.1 ± 3.0	⋯	${0.9}_{-0.2}^{+0.3}$	8.80	1.4 ± 0.2d	⋯	⋯
S244 5306	109.3434442	37.7458095	⋯	19.1 ± 3.0	⋯	${0.8}_{-0.6}^{+0.3}$	6.80	1.4 ±  0.2d,e	⋯	⋯
S39 6832	109.3626639	37.7313808	26.7 ± 8.6a	22.8 ± 3.2	⋯	${1.1}_{-0.1}^{+1.5}$	6.50	1.4 ± 0.2	⋯	⋯
S41 1901	109.3813749	37.7733429	⋯	⋯	22.6 ± 3.0	${1.02}_{-0.08}^{+0.07}$	6.00	1.7 ± 0.3	⋯	⋯
S44 2104	109.3766785	37.7714612	⋯	⋯	⋯	${0.82}_{-0.06}^{+0.06}$	6.50	1.6 ± 0.3	⋯	⋯
$S453811$	109.3919138	37.7583808	⋯	22.7 ± 2.6	⋯	${1.41}_{-0.19}^{+0.07}$	6.30	8.7 ± 4.1	⋯	⋯
$S484567$	109.3670705	37.7520604				${0.91}_{-0.06}^{+0.07}$	5.40	2.7 ± 0.9	⋯	⋯
S60 3367	109.3880803	37.7620638	⋯	8.3 ± 3.3a	⋯	${0.80}_{-0.05}^{+0.07}$	6.80	1.7 ± 0.2	⋯	⋯
$S{61}_{1}2016$	109.3770241	37.7723431	⋯	⋯	⋯	${1.73}_{-0.14}^{+0.09}$	7.60	3.4 ± 2.5d	⋯	⋯
$S{61}_{2}2015$	109.3769567	37.7724821	⋯	⋯	⋯	${1.6}_{-0.2}^{+0.2}$	8.30	3.4 ± 2.5d,f	⋯	⋯

Notes. The source ID and positions are taken from the CLASH catalog. The photometric redshifts (${z}_{{\rm{phot}}}$) and best fitting spectral template (t_b) are also taken from the CLASH catalog. The photometric redshifts are given at a 95% confidence level. The spectral templates are described in Molino et al. (2014). In total, there are 11 possible spectral templates, 5 for elliptical galaxies (1–5), 2 for spiral galaxies (6, 7), and 4 for starburst galaxies (8–11), along with emission lines and dust extinction. Non-integer values indicate interpolated templates between adjacent templates. For the radio flux density errors, we include a 2.5% uncertainty from bootstrapping the flux density scale. The listed amplification factors are the mean values from the models: CATS_v1 (Jauzac et al. 2012; Richard et al. 2014), Sharon_v2 (Johnson et al. 2014), Zitrin-ltm_v1, Zitrin-ltm-Gauss_v1 (e.g., Zitrin et al. 2013, 2009b), GLAFIC_v3 (Ishigaki et al. 2015), Williams_v1 (e.g., Liesenborgs et al. 2006), Bradac_v1 (Bradač et al. 2005, 2009), Merten_v1 (e.g., Merten et al. 2011). For the spectroscopic redshifts (${z}_{{\rm{spec}}}$), the spectral classification is as follows: A—absorption-line spectrum; E—emission-line spectrum.

^aManually measured. ^bFlux density measurement could be affected by radio emission from other sources. ^cVery faint source, only peak flux was measured. ^dThe complex morphology of the galaxy likely caused it to be fragmented and listed as separate objects in the catalog. ^eWe assume that these source components have the same redshift as S24₃. ^fWe assume that this source component has the same redshift as S61₁.

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Table 5.  Source Properties

Name/ID	R.A.	Decl.	${S}_{1.5}$	${S}_{3.0}$	${S}_{5.5}$	${z}_{{\rm{phot}}}$	tb	X-ray Flux	${z}_{{\rm{spec}}}$
	(°) (J2000)	(°) (J2000)	(μJy)	(μJy)	(μJy)			(10−15 erg cm−2 s−1)	Ebeling et al. (2014)
S1 3774	109.4073141	37.7583435	47.5 ± 11.2	22.6 ± 3.5	⋯	${0.46}_{-0.04}^{+0.07}$	6.80	⋯	⋯
S4 2833	109.4232228	37.7654146	105.7 ± 7.2	60.4 ± 2.2	24.7 ± 3.5	${0.29}_{-0.07}^{+0.06}$	5.90	⋯	⋯
S5 6341	109.4018584	37.7340578	442.2 ± 9.4	578.1 ± 3.2	446.2 ± 5.1	${0.69}_{-0.10}^{+0.04}$	5.50	⋯	0.5426 ± 0.0003 (E)
S7 5353	109.3982487	37.7457351	157.0 ± 9.2	57.6 ± 3.0	16.3 ± 3.5	${0.54}_{-0.02}^{+0.11}$	1.00	⋯	0.5408 ± 0.0003 (A)
S10 7521	109.3823125	37.7260450	64.4 ± 8.2	40.8 ± 2.5	19.1 ± 3.7	${0.56}_{-0.04}^{+0.06}$	6.60	⋯	0.5315 ± 0.0005 (E)
S11 7828	109.3816839	37.7225923	132.2 ± 9.9	85.3 ± 3.1	51.2 ± 4.3	${0.30}_{-0.07}^{+0.05}$	6.00	⋯	⋯
S13 7314	109.4023607	37.7268590	⋯	⋯	12.7 ± 5.0a	${0.57}_{-0.04}^{+0.07}$	8.20	⋯	0.5332 ± 0.0003 (E)
S14 1407	109.4268701	37.7779977	43.9 ± 13.5a	⋯	⋯	${0.6}_{-0.4}^{+3.0}$	7.90	⋯
S15 3462	109.4122491	37.7606844	35.7 ± 13.4a	⋯	⋯	${0.50}_{-0.04}^{+0.04}$	6.80	⋯	0.4992 ± 0.0003 (E)
S16 1096	109.4278576	37.7808548	66.2 ± 8.7	24.3 ± 3.1	26.9 ± 3.6	${1.0}_{-0.7}^{+3.1}$	8.50	⋯	⋯
S17 952	109.4095733	37.7806377	1685.9 ± 12.4	1048.8 ± 4.0	847.8 ± 7.9	${0.64}_{-0.05}^{+0.07}$	5.50	${5.64}_{-0.15}^{+0.18}$	0.5613 ± 0.0003 (A)
S20 7679	109.3522480	37.7248602	38.8 ± 8.7	25.4 ± 3.2	20.1 ± 6.0a	${0.9}_{-0.5}^{+3.4}$	7.90	⋯	⋯
S22 6064	109.3537889	37.7355689	26.7 ± 10.2a	16.5 ± 2.9	⋯	${0.49}_{-0.05}^{+0.04}$	6.00	⋯	0.5357 ± 0.0003 (E)
S23 5574	109.3647573	37.7448496	90.6 ± 9.3	48.6 ± 3.0	26.8 ± 3.2	${0.55}_{-0.04}^{+0.03}$	7.50	⋯	0.5261 ± 0.0003 (E)
S26 4123	109.3470266	37.7555482	85.0 ± 8.8	50.6 ± 3.2	27.0 ± 3.9	${0.5}_{-0.2}^{+0.2}$	6.50	⋯	0.4212 ± 0.0003 (E)
S27 3789	109.3478763	37.7578892	2013.0 ± 16.1	1274.5 ± 7.8	811.2 ± 9.2	${0.65}_{-0.37}^{+0.07}$	7.00	⋯	⋯
S28 3852	109.3739700	37.7568547	45.4 ± 10.1	24.6 ± 2.7	15.9 ± 3.5	${0.58}_{-0.03}^{+0.03}$	5.60	⋯	0.5565 ± 0.0003 (E)
S29 3402	109.3598798	37.7621055	⋯	19.7 ± 4.6a	17.0 ± 5.7a	${0.55}_{-0.03}^{+0.04}$	5.40	⋯	0.5442 ± 0.0003 (A)
S30 2446	109.3634032	37.7683109	33.5 ± 12.5a	31.5 ± 4.6	19.6 ± 3.2	${0.15}_{-0.05}^{+0.03}$	6.80	⋯	⋯
S31 1660	109.3568926	37.7756024	⋯	⋯	14.4 ± 5.8a	${0.9}_{-0.7}^{+2.9}$	7.90	⋯	⋯
S33 2974	109.3936629	37.7644686	⋯	⋯	⋯	${0.34}_{-0.03}^{+0.06}$	6.80	⋯
S34 4426	109.3981162	37.7514879	17797 ± 29	7058 ± 14	3149 ± 9	${0.56}_{-0.02}^{+0.03}$	5.50	⋯	0.5528 ± 0.0003 (A)
S35 4020	109.3985608	37.7547888	59.79 ± 5.1a,b	41.8 ± 8.0b	25.2 ± 2.4a,b	${0.54}_{-0.02}^{+0.02}$	4.30	⋯	0.5443 ± 0.0003 (A)
S36 5598	109.4050484	37.7397125	10505 ± 71	8356 ± 36	7132 ± 24	${0.16}_{-0.02}^{+0.06}$	1.30	${5.88}_{-0.14}^{+0.15}$	⋯
S38 6779	109.3951581	37.7315723	38.4 ± 4.9a,b,c	29.1 ± 6.7	29.5 ± 2.4a,b,c	${0.53}_{-0.05}^{+0.03}$	4.40	⋯	0.5366 ± 0.001 (A)
S42 5335	109.3813319	37.7437654	31.6 ± 11.2a	13.2 ± 3.1	⋯	${0.48}_{-0.03}^{+0.03}$	5.50	⋯	0.4919 ± 0.0003 (E)
S43 4992	109.3970833	37.7464913	⋯	19.8 ± 3.1	8.5 ± 3.3	${0.54}_{-0.03}^{+0.03}$	6.20	⋯	0.1779 ± 0.0003 (E)
S46 5274	109.4179222	37.7459331	36.3 ± 13.2a	23.0 ± 2.6		${0.50}_{-0.05}^{+0.04}$	9.40	⋯	0.5490 ± 0.0003 (E)
S47 5076	109.4220234	37.7478226	33.0 ± 12.3a	21.1 ± 6.8	⋯	${0.54}_{-0.05}^{+0.06}$	6.30	⋯	0.5660 ± 0.0003 (E)
S49 7693	109.3710599	37.7222104	⋯	7.5 ± 1.9a,c	⋯	${0.30}_{-0.18}^{+0.09}$	6.40	⋯	0.2288 ± 0.0003 (E)
S52 5299	109.3654725	37.7433269	105.0 ± 6.2	⋯	⋯	${0.22}_{-0.03}^{+0.03}$	7.00	⋯	⋯
S551d 1585	109.4246322	37.7763075	⋯	⋯	⋯	${0.4}_{-0.3}^{+2.5}$	8.00	⋯	⋯
S552d 1584	109.4245383	37.7763555	⋯	⋯	⋯	${2.5}_{-2.2}^{+0.6}$	7.90	⋯	⋯
S58 792	109.3976281	37.7831987	⋯	⋯	12.1 ± 5.2a	${0.8}_{-0.4}^{+1.1}$	7.60	⋯	⋯
S59 2444	109.3621323	37.7692569	⋯	13.5 ± 4.3a	⋯	${0.56}_{-0.05}^{+0.08}$	6.50	⋯	0.4984 ± 0.0005 (A)
S62 2977	109.3963967	37.7646847	⋯	19.9 ± 5.4a	⋯	${0.65}_{-0.05}^{+0.03}$	6.80	${3.09}_{-0.84}^{+0.98}$	0.5490 ± 0.0005 (E)
X63 1956	109.4139432	37.7728903	⋯	⋯	⋯	${0.50}_{-0.06}^{+0.03}$	5.20	${7.37}_{-2.87}^{+3.89}$	0.4897 ± 0.0003 (A)
X641d 2220	109.4099881	37.7711734	⋯	⋯	⋯	${0.8}_{-0.5}^{+2.4}$	8.00	${6.20}_{-4.71}^{+3.09}$	⋯
X642d 2222	109.4095458	37.7711695	⋯	⋯	⋯	${0.6}_{-0.2}^{+3.1}$	8.70	${6.20}_{-4.71}^{+3.09}$	⋯

Notes. The source ID and positions are taken from the CLASH catalog. The photometric redshifts (z_phot) and best fitting spectral template (t_b) are also taken from the CLASH catalog. The photometric redshifts are given at a 95% confidence level. The spectral templates are described in Molino et al. (2014). In total, there are 11 possible spectral templates, 5 for elliptical galaxies (1–5), 2 for spiral galaxies (6, 7), and 4 for starburst galaxies (8–11), along with emission lines and dust extinction. Non-integer values indicate interpolated templates between adjacent templates. For the radio flux density errors, we include a 2.5% uncertainty from bootstrapping the flux density scale. For the spectroscopic redshifts (z_spec), the spectral classification is as follows: A—absorption-line spectrum; E—emission-line spectrum.

^aManually measured. ^bFlux density measurement could be affected by radio emission from other sources. ^cVery faint source, only peak flux was measured. ^dThe complex morphology of the galaxy likely caused it to be fragmented and listed as separate objects in the catalog.

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To detect even fainter radio sources, we combined the individual L-, S-, and C-band images into one deep 1–6.5 GHz wideband continuum image, scaling with a spectral index²⁶ (α) of −0.5. This was done by convolving all maps to the resolution of the C-band image. With the help of this deep image, we identified about a dozen more sources below the PyBDSM detection threshold in the individual maps, but with peak fluxes above the local $3{\sigma }_{{\rm{rms}}}$. We manually determined the flux densities of these sources in AIPS with the task JMFIT. In addition, we visually identified five more sources in this deep image that were below the $3{\sigma }_{{\rm{rms}}}$ thresholds in the individual maps and thus not recognized there. These sources are also listed in Table 4 (and Table 5). We do not report integrated flux densities for these sources since they are not clearly detected in any of the individual maps. The CLASH photometric redshift and best fitting BPZ (Benítez 2000) spectral template for each source are also listed. For more details on the photometric redshifts and spectral templates, the reader is referred to Molino et al. (2014) and Jouvel et al. (2014). For 23 sources, spectroscopic redshifts are available from Ebeling et al. (2014).

For the sources that are located at redshifts larger than the cluster, we include the amplification factors in Table 4, taking the average over all the eight publicly available²⁷ HST Frontier Fields lensing models for MACS J0717.5+3745. The amplifications at a given redshift are calculated directly from the mass (κ) and shear (γ) maps. For more details on how these lensing models were derived, we refer to reader to the references provided in Table 4. The reported uncertainties in the amplification factors for the individual models are smaller than the scatter between these models. Therefore, for the uncertainty in the amplification factor, we take the standard deviation between these models, which should better reflect the actual uncertainties.

In total, we find 51 compact radio sources within the area covered by the HST imaging. In this sample, 16 sources are located behind the cluster, i.e., those where the 95% confidence limit of the lower redshift bound places it beyond the cluster redshift of z = 0.5458. Of these sources, seven have amplification factors larger than 2. We plot the location of the lensed radio and X-ray sources on top of an amplification map for a z = 2 source in Figure 3. We used the Zitrin-ltm-Gauss_v1 model amplification map as an example here. We discuss these lensed sources with amplification factors larger than 2 in some more detail in Section 3.2. About a dozen radio sources are associated with cluster members (i.e., $0.5\lt {z}_{{\rm{phot}}}\lt 0.6$).

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We also search for the presence of compact X-ray sources within the HST FOV. In total, we detect seven X-ray sources. Five of these have radio counterparts. The X-ray sources are also included in Table 4, with the measured X-ray fluxes. Two of these X-ray sources which have radio counterparts are lensed by the cluster. The other X-ray sources are foreground objects, cluster members, or have uncertain redshifts.

In this section, we discuss the radio and X-ray properties of the lensed sources with amplification factors $\gt 2$. These sources are double circled in Figure 1. For the sources that are not obvious AGNs (i.e., those that do not have X-ray counterparts), we compute the star-formation rate based on the measured radio luminosity. When converting from flux density to luminosity, we used the amplification factors listed in Table 4. The radio luminosity of the galaxies (non-AGN) can be converted to mean star-formation rate over the past $\sim {10}^{8}\;{\rm{years}}$ (Best et al. 2002) using

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{SFR}}}{{M}_{\odot }\;{{\rm{yr}}}^{-1}}\approx 4.5{\left(\displaystyle \frac{{\rm{GHz}}}{\nu }\right)}^{\alpha }\displaystyle \frac{{L}_{\nu }}{{10}^{22}\;{\rm{W}}\ {{\rm{Hz}}}^{-1}},\end{eqnarray} \tag{ 1 }$$

where L_ν represents the k-corrected rest-frame radio luminosity (which is also corrected for the amplification). The underlying assumptions are that cosmic rays from SNe II trace star-forming regions and that the number of SNe II is directly proportional to the SFR. An advantage of these radio-derived SFRs is that they are not significantly affected by dust extinction. For our sources, we use the 3 GHz measurement (unless stated otherwise), scaling with a spectral index of $\alpha =-0.5$. The uncertainty in the radio-derived SFR is about a factor of 2 (Bell 2003).

We can also compute the specific SFRs (sSFR) by computing the stellar mass (${M}_{\star }$) from the Spitzer 3.6 and 4.5 μm fluxes, following the approach by Rawle et al. (2014). We corrected these fluxes for the amplification and computed the K-correction using the BPZ spectral templates. The Spitzer fluxes are taken from SEIP Source List (Enhanced Imaging Products from the Spitzer Heritage Archive; we took the 38 diameter aperture flux densities). To compute the stellar mass, we use the relation from Eskew et al. (2012)

$$\begin{eqnarray}&&{M}_{*}[{M}_{\odot }]={10}^{5.65}{S}_{3.6\;\mu {\rm{m}}}^{2.85}{S}_{4.5\;\mu {\rm{m}}}^{-1.85}{({D}_{{\rm{L}}}/0.05)}^{2}{\rm{}}\;,\end{eqnarray} \tag{ 2 }$$

where S_λ is in units of Jy, and ${D}_{{\rm{L}}}$ the luminosity distance in megaparsecs. This relation was derived for the Large Magellanic Cloud and assumes a Salpeter (1955) initial mass function (IMF). It may break down for more strongly star-forming systems and might also vary with metallicity. Therefore, Equation (2) should be taken as an approximation to the stellar mass.

3.2.1. Comments on Individual Sources

It is expected that the number of star-forming galaxies increases with redshift, peaking at $z\sim 2$ (e.g., Madau & Dickinson 2014). We compute the radio luminosity function from the sources detected in our S-band image that have a 3 GHz flux density above 18 μJy ($10\times {\sigma }_{{\rm{rms}}}$). Above this flux density, we should be reasonably complete also for sources that are resolved.

For the luminosity function, we determine the volume behind the cluster in which we could detect a hypothetical source above the flux limit for a given luminosity. We then divide the number of detected sources (for that luminosity range) by the obtained volume. To compute the volume, we take the varying magnification (as a function of position and redshift) into account using the publicly available lensing models and provided Python code.

We excluded the regions covered by the radio relic. We scaled our 3 GHz S-band luminosities to 1.4 GHz taking α = −0.5, to facilitate a comparison with the literature results from Best et al. (2005). We limited ourselves to the range $0.6\lt z\lt 2.0$, because we only detect a single source above $z\gt 2$. Restricting the redshift range also limits the effects of the redshift evolution of the radio luminosity function.

The differences between the lensing models are larger than the uncertainties provided for the individual models. We therefore compute the volume for each for the eight models listed in Table 4 and take the average. The same is done for counting the number of sources in each luminosity bin (as this also depends on the amplification factors). In Figure 4 we plot the luminosity function averaged over these eight different lensing models. The red error bars represent the standard deviation over the eight lensing models. The black error bars show the combined uncertainty from the lensing models and Poisson errors on the galaxy number counts. These two errors were added in quadrature. We find that the uncertainties in the redshifts and flux density measurements do not contribute significantly to the error budget.

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Another uncertainty for the derived luminosity function is related to cosmic variance. Based on the computed volume, we probe for the three luminosity bins and the number of objects in each bin, we compute the cosmic variance using Trenti & Stiavelli (2008). From this computation, we find that cosmic variance introduces an extra uncertainty between 22% and 30%. Note, however, that this is a factor of a few smaller the Poisson errors and scatter due to the different lensing models.

In Figure 4 we also plot the low-redshift ($z\lt 0.3$) luminosity function derived by Best et al. (2005). Although the uncertainties in our luminosity function are substantial, we find evidence for an increase in the number density of sources of a factor between 4 and 10 compared to the Best et al. low-redshift sample.

With a larger sample (for example, including all six Frontier Fields clusters), it should become possible to map out the luminosity function more accurately, as the Poisson errors can be reduced by a factor of $\sim \sqrt{6}$. Surveys covering a larger area will be needed to map out the high-luminosity end. These surveys can be shallower and do not require the extra amplification by lensing. A major limitation of the precision that can be achieved for the faint end of the luminosity function, which can only be accessed with the power of lensing, is the accuracy of the lensing models. This highlights the importance of further improving the precision of the Frontier Fields lensing models (see also Limousin et al. 2015).