© The Authors 2023

1 Introduction

Radio sources with large extents in the sky may be not properly recovered by radio interferometers with an insufficient number of short baselines. A poorly sampled uv coverage at short spacings will thus cause unavoidable flux density losses (e.g. Wilner & Welch 1994; Deo & Kale 2017). This problem arises in many fields of radio astronomy, such as the study of extended radio galaxies, supernova remnants, and diffuse emission in galaxy clusters.

Among the various diffuse synchrotron sources found in galaxy clusters, radio halos are the most extended ones. They are centred on the cluster core region, extend up to megaparsec scales roughly following the distribution of the intra-cluster medium (ICM), and are characterised by steep (α ≳ 1) radio spectra¹ (see van Weeren et al. 2019 for a review). The origin of radio halos is associated with the re-acceleration of (primary or secondary) particles via stochastic Fermi II-type processes; these are driven by turbulence in the ICM induced by cluster mergers (Brunetti et al. 2001; Petrosian 2001; Brunetti & Lazarian 2007, 2016; Beresnyak et al. 2013; Miniati 2015). Nevertheless, the complex energy transfer mechanisms operating from large (~Mpc) to smaller scales are still unclear (see Brunetti & Jones 2014 for a review).

Radio halos are not ubiquitous in galaxy clusters. They are mainly found in massive² (M₅₀₀ ≳ 5 × 10¹⁴ M_⊙) and dynamically disturbed clusters, with increasing detection fraction with the host mass (e.g. Cassano et al. 2013; Kale et al. 2013; Cuciti et al. 2015, 2021b). The non-detections of radio halos may be intrinsically due to absence of radio emission (off-state clusters), or result from extrinsic instrumental and/or observational limits, namely a combination of insufficient sensitivity and poor sampling of short baselines.

In galaxy clusters where no radio emission is detected, upper limits on the radio power of a possible halo can be determined. Upper limits are important in statistical studies to constrain theoretical models of formation and evolution of the diffuse sources (e.g. Brunetti et al. 2007). In this respect, upper limits are necessary to understand whether off-state and on-state clusters belong to two distinct populations (relaxed and disturbed, respectively), and thus obtain constraints on the origin of the non-thermal emission. To this aim, it is necessary to compare the radio powers of detected halos and limits as a function of the mass of the host cluster (e.g. Cassano et al. 2013; Cuciti et al. 2021b). Moreover, deep upper limits can be exploited to test the level at which purely hadronic (Dennison 1980; Blasi & Colafrancesco 1999; Dolag & Enßlin 2000) radio halos may be detected or test models of dark matter interactions (Storm et al. 2017). Various methods have been adopted to determine upper limits. The most widely used method follows the ‘injection’ technique first exploited for radio halos by Brunetti et al. (2007) and Venturi et al. (2008). It consists of modelling simulated radio halos, whose predicted visibilities are added to the observed ones and then regularly processed to obtain images to check for possible detections at a given flux density threshold. Since then, this technique has been commonly adopted on data of galaxy clusters from different facilities (e.g. Kale et al. 2013; Bonafede et al. 2017; Johnston-Hollitt & Pratley 2017; Cuciti et al. 2021a; George et al. 2021a; Osinga et al. 2021; Duchesne et al. 2022).

The Low Frequency Array (LOFAR) is currently mapping the whole northern sky with unprecedented sensitivity and resolution at low frequencies through the LOFAR Two-Meter Sky Survey (LoTSS; Shimwell et al. 2017) with the High Band Antenna (HBA) operating at 120-168 MHz. Observations of galaxy clusters with LOFAR are promising to detect new radio halos, which are brighter at low frequencies owing to their steep synchrotron spectrum. The region of the sky covered by the Second Data Release of LoTSS (LoTSS-DR2; Shimwell et al. 2022) includes 309 galaxy clusters in the Second Planck Sunyaev-Zel’dovich (PSZ2) catalogue (Planck Collaboration XXVII2016), thus providing the largest sample of mass-selected clusters observed at low radio frequencies to date. This sample is extensively described in Botteon et al. (2022), showing the large variety of diffuse radio sources that were found³. In the present work, we focus on the sub-sample of 140 clusters where diffuse emission was not detected. By means of the injection technique, we generated simulated radio halos to test the capabilities of LOFAR to recover diffuse extended emission and obtain upper limits on the radio power of these clusters. These upper limits will be exploited in ongoing statistical analyses (Zhang et al. 2023, Cassano et al. 2023, and Cuciti et al., in prep.), where the properties of clusters with detected or undetected radio halos will be compared to investigate the origin of these populations.

This paper is organised as follows. In Sect. 2, we summarise the processes adopted to calibrate the LoTSS-DR2 data of Planck galaxy clusters that we will exploit in this work. In Sect. 3, we describe the procedures of injection. In Sect. 4, we simulate mock halos to test the capabilities of LOFAR to recover extended diffuse emission. In Sect. 5, we discuss the methods to obtain radio upper limits to non-detections of radio halos in galaxy clusters. In Sect. 6, we carry out simulations on other interferometers to be compared with the performances of LOFAR. In Sect. 7, we summarise our work. We adopted a standard ΛCDM cosmology with H₀ = 70 km s⁻¹ Mpc⁻¹, Ω_M = 0.30, and Ω_Λ = 0.70.

2 LoTSS-DR2 data

LoTSS-DR2 covers the 27% of the northern sky observed in the 120–168 MHz range (the nominal central frequency is 144 MHz). Each LoTSS pointing is 8 h long, and the typical resolution is ~6′′, and the median noise is ~0.08 mJy beam⁻¹ (Shimwell et al. 2022).

We refer the reader to Shimwell et al. (2019, 2022) and Tasse et al. (2021) for a complete description of the data processing by means of the Survey Key Project (SKP) pipelines. They include direction-independent and direction-dependent calibration and imaging through PREFACTOR (van Weeren et al. 2016; Williams et al. 2016; de Gasperin et al. 2019) and DDF-pipeline, which makes use of DDFacet (Tasse et al. 2018) and KillMS (Tasse 2014a,b; Smirnov & Tasse 2015), and finally the ‘extraction & re-calibration’ scheme (van Weeren et al. 2021) to further refine the quality of the images in the direction of the target. The final fully calibrated uv dataset only includes sources within the extraction region, and it can thus be easily manipulated in further analysis. Imaging is carried out by excluding baselines shorter than 80λ; these spacings are typically more challenging to calibrate and sample the possible emission on angular scales larger than ~40′ from our Galaxy that needs to be filtered.

In this work, we focused on a sample of 140 non-detections (NDEs) of diffuse emission from the ICM in Planck clusters belonging to LoTSS-DR2 (see Botteon et al. 2022 for details). According to our definition, diffuse emission from the ICM does not include emission associated with radio galaxies (e.g. lobes, filaments, tails, fossil bubbles) that may be present in the NDE clusters. We exploited these datasets to test the capabilities of LOFAR to recover extended emission and obtain upper limits to the presence of possible radio halos.

3 Injection algorithm

To test the capabilities of LOFAR and determine upper limits for the NDE clusters, we followed the injection⁴ technique. We first derived the visibilities from the Fourier inversion of a set of models of mock halos (‘prediction’ step) sampling a range of flux densities and angular sizes, and then added these mock visibilities to the dataset of each observation (‘injection’ step). Imaging and Fourier transforms were carried out by means of WSClean v. 2.10 (Offringa et al. 2014; Offringa & Smirnov 2017). The main steps of the procedure can be summarised as follows:

1. The coordinates (RA_inj ; Dec_inj) of the centre of the injection and the total injected flux density at 144 MHz (S _inj,tot) are required as inputs.

2. Frequency-dependent model images of the mock halo that follow an exponential surface brightness profile are built (see Sect. 3.1). We assumed a spectral index α = 1.3 in the 120–168 MHz frequency range, which is typical of radio halos. This is computed through the channels-out 6 parameter in WSClean, which produces model images at six sub-bands separated by 8 MHz each.

3. The model images are Fourier transformed by means of the predict function in WSClean to obtain the corresponding mock visibilities, which are then added to the real uv data.

4. The updated (real plus mock visibilities) datasets are imaged by adopting the multi-scale (fixed scales of [0, 4, 8, 16, 32, 64] pixels) and multi-frequency (channels-out 6) cleaning algorithm. To enhance the diffuse emission, the baselines are tapered by a Gaussian function (taper-gaussian) to lower resolutions.

An example of this process is shown in Fig. 1, where we injected a bright (S_inj,tot = 100 mJy) mock halo into a source-free region in the field of PSZ2 G059.18+32.91. To investigate the capabilities of LOFAR to recover extended emission and derive upper limits, we needed to repeat steps 2–4 by gradually decreasing S_inj,tot at step 1 until certain detection criteria (e.g. recovered flux density or sizes, signal-to-noise ratio, inspection by eye) are not fulfilled anymore, as discussed further in Sect. 5.2.

Dealing with LOFAR data is resource-intensive and requires proper computing and storage resources (e.g. Taffoni et al. 2022). We ran all the analyses on the HOTCAT High Performance Computing (HPC) cluster at INAF Trieste (Bertocco et al. 2020; Taffoni et al. 2020) by accessing nodes with 40 CPU cores and 6 GB of RAM each. Interacting with these nodes when running the pipelines is necessary to efficiently inspect intermediate results. To this aim, extra work in terms of setting up tunneling and virtual displays forwarding is required; moreover, advanced pipelines (such as those for LOFAR analyses) rely on complex dependencies which often require system-dependant setup steps. To overcome these issues, we made use of Rosetta (Russo et al. 2022), a container-centric science platform for interactive data analysis, which was recently made available for accessing the HOTCAT cluster. Rosetta can automatically set up interactive analysis environments (such as remote desktops and Jupyter Notebooks) on HPC cluster nodes by using software containers, thus allowing us to interactively access the cluster nodes and efficiently manage the dependencies of our pipelines.

Fig. 1 Example of injection process. The yellow circle is centred on (RAinj, Decinj) and has a radius of r = 3re. Left: model image at 144 MHz of the mock halo following the exponential surface brightness profile of Eq. (1). Middle: pre-injection image. Right: post-injection image.

3.1 Modelling of the surface brightness profile

Murgia et al. (2009) showed that a simple exponential law provides a good representation of the observed surface brightness distribution of a number of radio halos, with few free parameters. Following a common approach adopted for both real and mock halos (e.g. Murgia et al. 2009; Bonafede et al. 2017; Boxelaar et al. 2021; Osinga et al. 2021; Botteon et al. 2022; Hoang et al. 2022), we assumed a spherically-symmetric exponential law to model the surface brightness profiles of our mock halos:

$$I\left(r\right)=I_0{\text{e}}^{-\frac{r}{r\text{e}}},$$(1)

where I₀ is the central surface brightness and r_e is the e-folding radius.

The flux density is obtained by integrating Eq. (1) in circular annuli up to a certain radius $\widehat{r}$:

$$S_v=2\pi {\displaystyle {\int }_0^{\widehat{r}}I\left(r\right)r\text{d}r}=2\pi I_0\left[r_{\text{e}}^2-\left(\widehat{r}{r}_{\text{e}}+r_{\text{e}}^2\right){\text{e}}^{-\frac{\widehat{r}}{r_{\text{e}}}}\right].$$(2)

Equation (2) can be simply expressed as $S_v=2\pi f\left(\widehat{r}\right)I_0{r}_{\text{e}}^2$, where $f\left(\widehat{r}\right)=\left[r_{\text{e}}^2-\left(\widehat{r}{r}_{\text{e}}+r_{\text{e}}^2\right)e^{-\frac{\widehat{r}}{r_{\text{e}}}}\right]$ is the fraction of flux density within the integration radius r̂ to the total one (f (r̂) = 1 when r̂ = +∞). Radio halos do not extend indefinitely, and thus their emission is typically measured up to r̂ = 3r_e, which provides a fraction f(r̂) = 0.8 of the total flux density.

For the injection process, we derive the central brightness from the total injected flux density as $I_0=S_{\text{inj,tot}}/2\pi r_{\text{e}}^2$. In the following, we refer to the injected diameter of the mock halo as being D = 6r_e (i.e. we assume a radius R = 3r_e).

3.2 Schemes of injection

As examples, in Figs. 2, 3 we show the images before and after the injections of mock halos with r_e = 40.7′′ and S_inj,tot = 20, 50, and 100 mJy that we performed both close to the pointing centre of PSZ2 G120.08-44.41 and far from it (at a distance of ~19′) using the extracted and re-calibrated datasets (‘Inj.’ scheme). As preliminary checks of our procedure, we fitted the surface brightness of the mock halos (discrete sources close to the mock emission were masked) by means of the Halo-FDCA⁵ code (Boxelaar et al. 2021), which was used for radio halos in LoTSS as well (van Weeren et al. 2021; Botteon et al. 2022; Hoang et al. 2022), and we measured the flux density in circles of radius r̂ = 3r_e, where a fraction f(r̂) = 0.80 of the total injected flux density is expected according to Eq. (2). The images in Fig. 3 show that injecting the same flux density in different locations generally provides visually different mock halos because of the presence of discrete sources and local noise fluctuations, contaminating the morphology of the mock emission. Nevertheless, as reported in Table 1, the flux densities measured by hand (S_meas) are in good agreement for all the corresponding injections in the centre and far from it. The flux density obtained through Halo-FDCA is computed as $S_{\text{fit}}\propto I_{0,\text{fit}}{r}_{\text{e,fit}}^2$, and thus it is strongly dependent on the fitted e-folding radius and its associated error; in particular, the less significant the emission of mock or real halos, the higher the errors on r_{e, fit}. Despite this, all the corresponding injected and fitted parameters are consistent within the fitting errors. These tests indicate that our simulations are barely dependent on the position of the injection, meaning that the response of the instrument can be considered uniform, at least across the extraction regions.

Background, foreground, and embedded sources contaminate the faint diffuse emission of radio halos. A model of the discrete sources can be obtained by selecting only the longest baselines, depending on the angular scales that need to be filtered out, and their contribution can be then directly subtracted from the uv data. We compared the results of the ‘Inj.’ scheme with the injection in source-subtracted datasets (‘Sub. & Inj.’ scheme) and the effects of subtraction after injecting in the original datasets (‘Inj. & Sub.’ scheme). To this aim, following the approach described in Botteon et al. (2022), we selected the baselines corresponding to projected sizes <250 kpc at the cluster redshift, and removed discrete sources in our targets. As reported in Table 1, the measured and fitted flux densities are consistent for all the three schemes. We notice, however, differences in I_0,fit and r_{e, fit} in case of the injection with the lowest flux density, that is S_inj,tot = 20 mJy. As previously mentioned, this is due to the low significance of the diffuse emission with respect to the background, and in these cases, residuals from the source subtraction process that were not masked during the fit could have a non-negligible impact on the fit with Halo-FDCA. By comparing the ‘Sub. & Inj.’ and ‘Inj. & Sub.’ schemes, we find an almost perfect agreement in both the measures and fitting of the corresponding cases, meaning that the mock diffuse emission is not included in the subtraction model. This latter result has a major practical utility for our work; the subtraction can be performed only once, thus allowing us to save a huge amount of computing time, and different injections can be carried out in the same source-subtracted dataset, with no need to subtract sources after each injection (see also Sect. 5.2).

We conclude this section by commenting on the recovered morphology of mock halos. For very bright mock halos, the spherical and smooth profile is typically recovered. On the other hand, in the case of less bright mock halos, the roundish and smooth shape can be easily perturbed depending on the local noise pattern and effective sensitivity of the observations. By gradually decreasing the injected flux density, the symmetry is progressively broken, and the mock halo will appear as patches of emission around the peak. This behaviour can be seen in Fig. 3. Even though the recovered morphology depends on the specific dataset and injection position, we can quantify the relative deviations as a function of radius and injected flux density through a point-to-point comparison. To this aim, we considered our brightest injections at 100 mJy (with the ‘Sub. & Inj.’ scheme), and sampled the corresponding mock halos with a grid of beam-size square boxes down to the 2σ contour level. The same grid was used to sample the mock halos of S_{inj, tot} = 50 and 20 mJy as well. The surface brightnesses at 20 and 50 mJy were normalised to match the injection at 100 mJy (i.e. they were multiplied by a factor of 5 and 2, respectively), and the corresponding measures were reported in Fig. 4 as a function of the brightness at 100 mJy. These plots show that progressively increasing deviations from the one-to-one line are found from higher to lower brightness (i.e. outwards from the peak) and that the scatter is larger for the injections at 20 mJy than at 50 mJy. This can be also observed in the insets of Fig. 4: the reported radial profiles are obtained by averaging profiles extracted in four 90°-wide sectors for each mock halo. Both the point-to-point plot and radial profiles show that the scatter of the brightness is larger in the outermost regions of the mock halos, where the faint emission consists of asymmetric patches and less flux density is recovered. Moreover, the differences between the injections in the centre and periphery are associated with the local noise conditions, which notably contribute to the recovered morphology of the mock halo. By decreasing S_{inj, tot} and approaching the noise level, the scatter will further increase.

Fig. 2 Restored images before injections in centre (left panels) and periphery (right panels) of PSZ2 G120.08-44.41 before (upper panels) and after (lower panels) subtraction of discrete sources.

Fig. 3 Restored images after injections in centre (panels A–C) and periphery (panels D–F) of PSZ2 G120.08-44.41 with different schemes: injection in the original dataset (‘INJ.’, panels A and D), injection in the discrete source-subtracted dataset (‘SUB. & INJ.’, panels B and E), and injection in the original dataset and subsequent subtraction of the discrete sources (‘INJ. & SUB.’, panels C and F). The yellow circle (centred on RAinj, Decinj) has a radius of 3re and contains Sinj = 0.8Sinj,tot, where Sinj,tot = 100, 50, and 20 mJy, for panels in Cols. 1–3, respectively.

Fig. 4 Point-to-point comparison of surface brightness of mock halos in PSZ2 G120.08-44.41 (‘Sub. & Inj.’ scheme; central and peripheral injections are shown in the upper and lower panel, respectively). Values at 20 and 50 mJy are multiplied by a factor of 5 and 2, respectively, to match the injection at 100 mJy. The one-to-one line is plotted in black. Insets show the corresponding radial profiles, whose points are obtained as the average brightness of four 90°-wide sectors, whereas the shaded region represents their standard deviation. The point-to-point plots show that the lower the brightness and the injected flux density, the higher the scatter around the one-to-one line, indicating the progressive apparition of emission patches. Analogously, the radial profiles indicate that the scatter increases at larger radii and lower brightness.

4 LOFAR performances

In this section, we explore LOFAR’s ability to recover extended emission in detail, by simulating mock radio halos with a wide range of flux densities and angular sizes in some clusters of our reference sample. By adopting the standard uv cut for LOFAR HBA, baselines <80λ will be not employed in our analysis.

4.1 The role of the uv coverage

When dealing with extended radio emission, the role played by the uv coverage needs to be carefully taken into account. Indeed, interferometers lacking short baselines will not be able to fully recover extended sources in the sky. As a consequence, non-detections might arise from insufficient sampling of the uv coverage at short spacings. In previous works, LOFAR uv coverage was tested on the basis of injections on specific observations (Hoang et al. 2018; Botteon et al. 2020). Here, we aim to systematically investigate the associated flux density losses by performing ~500 injections in ~10 cluster fields characterised by different data quality and noise. In each of these clusters, we injected mock halos with various flux densities (ranging from 5 to 145 mJy) and angular diameters (3, 5, 7.5, 10.5, 14, 18arcmin, corresponding to baseline lengths of ~1150kλ, 690λ, 460λ, 330λ, 250λ, 190λ, respectively). As a representative example of one of these clusters, in Appendix A we show a collection of mock halos injected in the field of PSZ2 G098.62+51.76, which was used to produce the plots discussed throughout this and next section. For each mock halo, we obtained the corresponding azimuthally averaged surface brightness profile I_post(r) by adopting sampling annuli of widths equal to half of the full width at half maximum (FWHM) of the restoring beam of the images. The same analysis was performed on the pre-injection images, whose profiles I_pre(r) include the contribution of both radio sources and noise.

In Fig. 5, we show the net profiles computed as the difference in each bin between I_post and I_pre, overlaid on the injected profiles after being convolved with the restoring beam; since the peak of the profiles is always recovered, the first sampling annulus is not reported. The recovered net profiles are in agreement with the injected profiles for mock halos with sizes up to D = 10.5′ ; this result is a consequence of the dense uv coverage of LOFAR on short spacings. On the other hand, deviations are visible in the cases of D = 14′ and D = 18′.

To estimate the losses in cases of extended sources, in Fig. 6 we report the fraction of recovered flux density as a function of the injected flux density for each angular scale D ≥ 7.5′. Overall, the recovered fractions are approximately constant over a wide range of injected flux densities up to D = 14′ .To further constrain the effective losses, we inspected the images shown in Appendix A and determined the threshold among the injected flux densities S_inj at which the mock emission can no longer be distinguished from the local noise. Based on these flux density levels (indicated by triangular markers in Fig. 6), on average flux density losses associated with the uv coverage are <5% up to D = 10.5′, ~10% for D = 14′, and >20% for sources with D = 18′. In summary, we conclude that flux density measurements of sources observed by LOFAR are weakly dependent on the uv coverage up to D ~ 15′, meaning that the instrument is still well sampled at spacings ~250λ, whereas non-negligible losses should be taken into account for sources with larger angular sizes. In the latter case, the use of baselines <80λ should be considered to mitigate the flux density losses.

It is worth mentioning that the recovered net profiles can appear higher than the injected profiles. This effect is described in detail in Shimwell et al. (2022) and is due to uncleaned components during the imaging step: when the peak of a cleaning component is lower than the cleaning threshold, it will be not correctly deconvolved and will be enhanced in the restored image. As seen in Fig. 5, this cleaning bias increases for wider (large r_e) and fainter (low I₀) profiles. In the worst cases, such as S_irj,tot = 5 mJy, the recovered peaks are biased by factors of ~1.2, 1.4, 1.6, 1.6, and 2, for D = 5′, 7.5′, 10.5′, 14′, and 18′, respectively, in agreement with Shimwell et al. (2022), who also found maximum factors of 2 through injections of Gaussian profiles. For a comparison, with S_inj,tot = 10mJy and D = 18′, the maximum cleaning bias factor is ~1.5, and it further decreases for higher injected flux densities. Therefore, even if this systematic effect is present, the global results we draw in this section still hold.

This work focuses on exponential profiles as they can reproduce the observed brightness of radio halos, but losses with different injected models can be derived with the same approach. Shimwell et al. (2022) probed the fraction of recovered flux density in LoTSS-DR2 by injecting Gaussian profiles, and found that this is ~95% for standard deviations ≤2′. Gaussian profiles are narrower than exponential profiles of the same width. Therefore, higher flux density fractions are expected to be recovered because of the increasing contribution of longer baselines. A systematic comparison between exponential and Gaussian profiles is beyond the aim of the present paper. However, as a sanity check, we injected bi-dimensional Gaussian profiles in the form of $I\left(r\right)=I_0{\text{e}}^{-\frac{r^2}{2r_{\text{e}}^2}}$ (where r_e now represents the standard deviation of the Gaussian and $S_{\text{inj,tot}}=2\pi I_0{r}_{\text{e}}^2$in PSZ2 G098.62+51.76. We found losses ≲ 10% up to D = 18′ (i.e. r_e = 3′), in agreement with results of Shimwell et al. (2022).

Fig. 5 Net azimuthally averaged surface brightness profiles of the images shown in Appendix A for PSZ2 G098.62+51.76. The dashed black lines represent the theoretical injected profiles. The sampled profiles (Sinj, tot is reported in the legend) are obtained by subtracting the pre-injection from the post-injection contribution. The grey vertical line indicates r = 3re.

Fig. 6 Fraction of recovered net flux density as a function of the injected flux density (within 3re) at various angular scales. Triangular markers indicate the flux density threshold at which the mock emission is not visible based on images in Appendix A. Non-negligible flux density losses (>15%) are expected for sources with angular sizes D > 15′ only.

4.2 Dependencies of the upper limits

Through our simulations, limited to baselines >80λ only, we found that flux density losses are negligible and we can consider both detections and non-detections of extended emission independent on the uv coverage of LOFAR up to D ~ 15′ (see also Sect. 6 for a comparison with other facilities). Therefore, we expect non-detections of radio halos in clusters to be intrinsic or depend only on the sensitivity due to the depth of a specific observation. As a consequence, this suggests that our upper limits should rely on stringent parameters, which we aim to determine.

In Fig. 7, we show both the post-injection (I_post(r)) and pre-injection (I_pre(r)) surface brightness profiles (bright discrete sources were masked for this step). Uncertainties on the flux density of each sampling annulus are

$$\text{Δ}{S}_{\text{\hspace{0.17em}bin}}=\sigma \sqrt{N_{\text{beam,bin}}},$$(3)

where σ is the noise (in beam area units) of the radio image and N_{beam, bin} is the number of beams within each annulus. Therefore, the reported errors in Fig. 7 are computed as follows:

$$\text{Δ}{I}_{\text{bin}}=\frac{\text{Δ}{S}_{\text{\hspace{0.17em}bin}}}{N_{\text{beam,bin}}}=\frac{\sigma }{\sqrt{N_{\text{beam,bin}}}}.$$(4)

We inspected our images, and, as a rule of thumb, we found that the mock halos are still visible if at least two bins (as mentioned, the peak is excluded) are above the local noise level (i.e. I_post > I_pre) in our plots. Otherwise, the mock emission cannot be distinguished from the noise, allowing us to define the upper limit in combination with visual inspection.

For each set of injections, we determined the upper limit by eye, guided by the azimuthally-averaged surface brightness profiles. We found that limits depend on a combination of the noise (σ) and resolution (θ) of the restored image, and on the angular size of the mock halo. In particular, the flux density of the upper limit correlates with the noise and the number of beams N_beam ~ D²/θ² within the injected mock halo. As shown in Fig. 8, we performed a simple linear regression of the points (~50 upper limits among our injections) in a logarithmic plane as

$$\log \left(\frac{S_{\text{\hspace{0.17em}UL}}}{\sigma }\right)=m\log \left(N_{\text{beam}}\right)+q,$$(5)

$$P_{\text{UL,150}}=0.8\times f_{\text{LoTSS}}\times 4\pi D_{\text{L}}^2{S}_{\text{\hspace{0.17em}UL,150}}{\left(1+z\right)}^{\alpha -1},$$(6)

where m and q are the fitted slope and intercept, respectively.

Possible faint diffuse emission can be more easily identified by visual inspection if it is spread over larger areas, meaning that when N_beam is large, upper limits are guided by a surface brightness criterion; conversely, they follow a flux density criterion for smaller N_beam. If our limits were determined based on a surface brightness criterion only, the flux density would scale with the area of the source as $S\propto r_{\text{e}}^2\propto N_{\text{beam}}$, and thus we would expect a slope of m = 1 in Eq. (5), whereas a flatter slope of m = 0.5 is expected if limits are exclusively driven by a flux density criterion (see also Fig. 3 in Brunetti et al. 2007).

These two regimes can be observed in Fig. 8, with a flattening that roughly occurs for N_beam < 50. We performed different linear regressions by using all the points (blue line), points with N_beam > 50 (red line), and points with N_beam < 50 (green line); the results of these fits are reported in Table 2. The average slope m = 0.703 that we obtained by fitting all the points is in line with the competing trends predicted by the two criteria. By considering only points with larger N_beam, the fitted slope m = 0.789 steepens, in agreement with the behaviour predicted by the surface brightness criterion. We notice that the same (red) line can interpolate points down to N_beam ~ 30 as well. Indeed, by considering points with N_beam > 30 we obtained a similar slope m = 0.794, which is consistent with that obtained with N_beam > 50 within the fitting errors.

For fixed σ and D, this positive correlation shows that deeper upper limits can be obtained for injections of lower N_beam, which are achievable by decreasing the resolution of the restored image. Even though worse σ are obtained by tapering the baselines, we found that images tapered to resolutions of 90′′ have typical rms, which are factors ≲2.5 only with respect to those of images tapered to resolutions of 30′′. According to these results, the depth of the limit is primarily driven by N_beam ∝ θ⁻² and it generally benefits from the decrease of the resolution.

Fig. 7 Azimuthally averaged profiles of images shown in Appendix A for PSZ2 G098.62+51.76. The dashed black lines represent the theoretical injected profiles. The sampled pre-injection and post-injection profiles are shown with black and coloured dots (see the total injected flux density in the legend), respectively. The grey vertical line indicates r = 3re. The grey horizontal line indicates the global 1σ noise level.

Fig. 8 Logarithmic plot of the ratio of upper limit flux density against the noise of the map (SUL/σ) as a function of the number of beams (Nbeam) within the injected halo. The blue, red, and green lines are the fitted linear regressions (see Table 2) obtained with no cut in Nbeam, with Nbeam > 50 and with Nbeam < 50, respectively. The grey vertical lines are drawn at Nbeam = 30 and Nbeam = 50.

5 Upper limits for PSZ2 clusters in LoTSS-DR2

In this section, we describe the procedures adopted to obtain the upper limits for the NDE clusters of our sample, and compare them with the flux densities of the detected radio halos in LoTSS-DR2.

5.1 NDE cluster sample

From our initial sample of 140 NDE Planck clusters, we first excluded 26 objects whose redshift is unknown, thus not allowing us to make assumptions on the angular size and limits to the radio power of a possible halo. For 11 additional galaxy clusters, upper limits would not be reliable due to the presence of contaminating AGN with extended emission and/or calibration artefacts in their central regions. Finally, we did not consider the lowest-redshift NDE cluster (PSZ2 G136.64-25.03, z = 0.016) because a possible radio halo of ~1 Mpc would have an angular size of ~1° at the cluster redshift; sources of such large angular sizes require more specific calibration procedures than those adopted in LoTSS, making use of all the baselines instead of restricting to those > 80λ (see e.g. the case of the Coma cluster in Bonafede et al. 2021). Therefore, we focus on a sample of 102 NDE galaxy clusters. As shown in the upper panel of Fig. 9, these clusters lie in the redshift range [0.062–0.9], where the mean and median are z̄ = 0.318 and z͂ = 0.267, respectively. We refer the reader to Table 1 in Botteon et al. (2022) for additional information (M₅₀₀, R₅₀₀, image quality, X-ray data availability) on the full NDE cluster sample.

Fig. 9 Distributions of redshifts and angular diameters of the 102 NDE galaxy clusters in our final sample. Top: distribution of redshifts. Bottom: distribution of the injected angular diameters under the assumption of re,inj = 200 kpc.

5.2 Upper limit calculation

The injection technique requires input values of I₀ and r_e. As mentioned, the central brightness is obtained as $I_0=S_{\text{inj,tot}}/2\pi r_e^2$. In past works, the e-folding radius was derived from scaling relations with the host cluster mass, but these are still poorly constrained and should be used with caution, as discussed in Bonafede et al. (2017). To avoid the use of scaling relations, we consider mass-independent e-folding radii.

As discussed in Sects. 4.1 and 4.2, our upper limits depend on the extent (in terms of number of beams) of the mock halo and the noise of the image, while the role of the uυ coverage is negligible for the extension of halos in our LoTSS sample. Therefore, in principle, by means of Eq. (5), we are able to immediately calculate the upper limits for the 102 NDE galaxy clusters of our sample as $S_{\text{UL}}={10}^q\sigma N_{\text{beam}}^m$ (with m = 0.789 and q = −0.091) without performing any additional injection. To this aim, values of σ and N_beam are required, which can be obtained from images of the original datasets and with a choice of r_e. Bearing in mind that the mean and median e-folding radius of the PSZ2 radio halos in LoTSS-DR2 are r̄_e = 194 kpc (with a standard deviation of 94 kpc) and r͂_e = 186 kpc, respectively, we assumed a nominal r_e = 200 kpc for all the NDE objects to derive the corresponding angular diameter (D = 6r_e). With this choice, the sizes of our mock halos are in the [2.6′–16.7′] range. The distribution of the angular diameters is shown in the lower panel of Fig. 9, where the mean and median are D̄ = 5.7′ and D͂ = 4.9′, respectively; 82% of our clusters have D < 7.5′, whereas only two clusters have D > 14′. We then produced images by tapering the baselines to different convenient low resolutions to find the deepest S_UL from Eq. (5) among the various combinations of σ and N_beam, under the condition of avoiding N_beam ≪ 30, where the slope of our correlation significantly flattens.

Having said that, since radio halos are centrally located sources, physically-meaningful upper limits cannot ignore the local environment close to the cluster centre, which is usually dense in contaminating discrete sources. Therefore, at this stage it is also necessary to take into account the quality of the subtraction of the discrete sources close to the cluster centre, before blindly adopting Eq. (5). The subtraction process is typically not perfect for many reasons; it assumes that the model of discrete sources obtained from the long baselines adequately describes their emission for the short baselines as well, but extended sources are primarily sampled by the short spacings, and calibration may not be homogeneous for all baselines, thus providing unequal levels of subtraction. These effects may introduce subtraction artefacts in the form of positive residual blobs and negative holes, which are enhanced at low resolution and contaminate the faint diffuse emission; in these cases, upper limits are not driven by the instrumental capabilities (i.e. the reached rms noise), but by the level of imaging artefacts, thus making the S_UL provided by Eq. (5) not fully trustworthy.

For these reasons, we inspected our non-subtracted and source-subtracted images at various resolutions and excluded 27 out of 102 objects affected by severe subtraction artefacts that prevent us from providing meaningful upper limits from our analysis⁶ (see an example in Fig. 10). We then assigned a subtraction-quality (SQ) parameter to each remaining NDE cluster, based on the presence and impact of subtraction artefacts close to the cluster centre⁷; we assigned SQ = 1 if subtraction artefacts are absent or negligible (a sub-sample of these targets was used to derive Eq. (5)) and SQ = 2 if subtraction artefacts are not negligible (see examples in Fig. 10) . We directly derived upper limits through Eq. (5) only if SQ = 1 (15 out of 102). If SQ = 2 (60 out of 102), the presence of artefacts is not dominant and more reliable limits can be determined through the injection process in the source-subtracted data (i.e. through the ‘Sub. & Inj.’ scheme). In these cases, we performed a first cycle of injection by using the flux density provided by Eq. (5) with tapers corresponding to N_beam ~ 30, and then increased or decreased S_inj,tot in few additional cycles to further constrain the limit level. Pre-injection and post-injection images were then inspected and, guided by the 2σ contour levels, we considered cases where the mock emission leaves extended excess ~2 times brighter than the previous injection cycle as effective limits (see an example in Fig. 11).

Based on the scatter around the fit of Eq. (5) in the region N_beam ~ 30–50, and the grid of values adopted to vary S_inj,tot in each cycle of injections, we claim conservative uncertainties of ~10–15% on the upper limits derived with the presented methods. As discussed in Sect. 4.1, the surface brightness is overestimated if emission is not fully deconvolved during imaging. This cleaning bias is expected to be higher for injections at the level of the upper limit and large angular size. Nevertheless, even in the case we were systematically biased by this cleaning effect, uncleaned excess would appear brighter, thus making the upper limits to be more conservative.

We now aim to evaluate the efficacy of our strategies based on Eq. (5). For each of the 60 NDE clusters with SQ = 2, we obtained an estimate of the expected upper limit from Eq. (5) by assuming N_beam = 30 and the noise corresponding to the adopted taper. In Fig. 12, we report the distribution of the ratio between the effective and expected upper limits; the mean and median ratios are 1.59 and 1.44, respectively. Most of the targets (51 out of 60) have ratios of ≤2, with maximum ratios of ~3.3. The nine clusters with ratios of > 2 have slightly higher levels of artefacts than those typical of SQ = 2, and therefore they require higher injected flux densities to constrain the upper limit. This confirms that Eq. (5) remains a good starting point even in the presence of moderate artefacts, thus allowing us to perform fewer additional cycles of injections in order to constrain the final level of the limit. Moreover, even though the required time to obtain upper limits naturally depends on the available computing resources (see details in Sect. 3), we estimate that our strategies allowed us to greatly reduce the computing time by factors of ~3–m,4⁸ with respect to standard methods in the literature.

Fig. 10 Representative examples of NDE clusters showing artefacts introduced by the subtraction of discrete sources near the cluster centre. The radio contours are drawn at ±2σ and spaced by factors of 2. In all the panels, the dashed cyan circle indicates the cluster centre and has a fixed diameter of 2′. Left: NDE cluster excluded from our analysis due to severe (positive and/or negative) artefacts. Middle: NDE cluster with SQ = 1 due to the absence of artefacts. Right: NDE cluster with SQ = 2 due to the presence of moderate residuals.

Fig. 11 Examples of injections for SQ = 2 cluster. Contours are drawn at 2σ of the pre-injection (left) image, and the cyan circle has a radius of 3re,inj. Two cycles of injections are performed with SUL derived from Eq. (5) and 1.5 × SUL. The upper limit is obtained with the second cycle, which leaves extended excess ~2 times brighter with respect to the pre-injection image.

Fig. 12 Distribution of the ratio between the effective upper limit and the expected upper limit inferred from Eq. (5) for the SQ = 2 clusters.

Fig. 13 Distributions of flux density of confirmed and candidate radio halos (in blue) with 100 ≤ re ≤ 400 kpc from Botteon et al. (2022), our upper limits (in red), and upper limits from Cuciti et al. (2021a; in black), and George et al. (2021a; in yellow). Upper limits from the literature are re-scaled to 150 MHz by assuming α = 1.3. The samples are split in redshift bins as z < 0.3 (top panel) and z > 0.3 (lower panel).

5.3 Upper limits and radio halos

Uncertainties in the beam model of LOFAR HBA can introduce offsets in the flux-density scale when amplitude solutions are transferred from the primary calibrator to the target (e.g. Hardcastle et al. 2016). Therefore, all images need to be multiplied by a factor of f_LoTSS (of the order of unity), which is derived for each pointing after the data calibration (see Botteon et al. 2022; Shimwell et al. 2022) to align the flux density scale of LoTSS with that of Roger et al. (1973). As our injections are performed in the uv data, we have to take into account the flux density scale correction by multiplying S_UL at 144 MHz by f_LoTSS.

Our upper limits will be exploited in statistical analysis in other papers of the series (Zhang et al. 2023; Cassano et al. 2023, and Cuciti et al., in prep.), where the radio power of the detected halos is reported at 150 MHz, within radii of 3r_e (as done in Botteon et al. 2022). To compare detections and upper limits, we therefore scaled S_UL from 144 to 150 MHz (S_UL,₁₅₀ = S_UL,144(150/144)^−α, where α = 1.3), calculated the corresponding radio power, and finally considered the 80% of this value which is expected within 3r_e,inj. In summary, the k-corrected radio powers of the upper limits are obtained as

where D_L is the luminosity distance at the cluster redshift.

The main host properties (Planck coordinates, redshift, mass), injection parameters (centre of injection, central brightness, e-folding radius), flux density, and radio power at 150 MHz for the final sample of 75 NDE clusters are reported in Table B.1.

A comparison of the distributions of flux densities for our upper limits and the 56 (confirmed and candidate) radio halos detected in LoTSS-DR2 having 100 ≤ r_e ≤ 400 kpc (Botteon et al. 2022) is shown in Fig. 13, where we also report upper limits from the literature (Cuciti et al. 2021a; George et al. 2021b) that we re-scaled to 150 MHz. For a clearer inspection, we split the samples in two redshift bins at z < 0.3 (upper panel) and z > 0.3 (lower panel). Our limits range from 1.5–17 mJy, whereas radio halos have flux densities approximately in the 3–700 mJy range. Except for a few objects, the distributions of our limits are well separated from the radio halos in the lower redshift bin, whereas more overlapping is found in the higher redshift bin. Our limits are deeper than those from the literature obtained with other interferometers, thanks to the higher sensitivity of LOFAR (see also Sect. 6). The deepest limits at 150 MHz are 36.5 mJy (at z = 0.320) and 11.3 mJy (at z = 0.396) in samples from Cuciti et al. (2021b) and George et al. (2021b), respectively, which are factors of 24.3 and 7.5 higher than our deepest limit of 1.5 mJy (at z = 0.830), and factors of 2.8 and 6.5 higher than our limits for objects at similar redshifts (z = 0.397 and z = 0.318).

6 Comparison with uGMRT and JVLA capabilities

To date, studies of radio halos have been mainly based on observations carried out with the upgraded Giant Metrewave Radio Telescope (uGMRT) and the Karl G. Jansky Very Large Array (JVLA). To quantitatively compare the capabilities of these instruments with LOFAR, we injected mock halos with S_inj,tot ranging from 20 to 300 mJy in LOFAR (8 h observation), uGMRT (band 3 at 400 MHz, 6 and 10 h observations), uGMRT (band 4 at 700 MHz, 9 h observation), and JVLA (L band in combined DnC and BnC array configurations at 1.5 GHz, 8 hr on source in total) datasets. Even though the minimum baseline lengths of uGMRT and JVLA are similar to the inner uv cut considered for LOFAR at 80λ, the density of the uv coverage at short spacings is notably different for each dataset.

As in Fig. 6, we obtained the recovered flux density by considering the difference between S_post and S_pre. Sources were not subtracted to avoid the introduction of heterogeneous artefacts of subtraction at the various frequencies. In Fig. 14, we show the fraction of the recovered flux density as a function of the injected flux density for the three considered instruments. Losses are negligible for all three facilities up to D = 5′ (and thus D = 3′ and D = 5′ are not shown), but different behaviours are seen for larger angular sizes. To roughly determine the effective losses, we consider a representative radio halo of D = 1.2 Mpc, host mass of M₅₀₀ = 5 × 10¹⁴ M_⊙, and spectral index α = 1.3; we infer its radio power at 150 MHz from the mass-power relation (with fitted slope and intercept from Cuciti et al., in prep.) and re-scale its flux density based on the required angular diameters and frequency; the corresponding flux density is indicated by a vertical dashed line in Fig. 14. Our estimates on the effective recovered fractions are listed in Table 3 (due to large differences, for uGMRT band 3 we reported both the minimum and maximum values, obtained with 6 and 10 h observations, respectively).

Obtaining more accurate assessment of the performances for uGMRT and JVLA requires systematic injections in many datasets, as done for LOFAR, but this is beyond the scope of this work. However, important conclusions can be drawn through our simulations. Owing to the high number of short baselines of LOFAR, the 8 hr LoTSS pointings ensure similar densities of the uv coverage, independently of the specific observation. We found that uGMRT can provide very high performance levels in recovering extended emission, but the recovered trends are remarkably different for the uGMRT datasets that we considered. Indeed, Figs. 14 and 15 indicate that the density of uGMRT uv coverage is more dependent on the specific observation (e.g. total observing time, bandwidth, declination of the target, level of interference, flagging); in our tests, the lowest performances were obtained with the 6 hr band-3 dataset, which has not only the shortest duration, but also the highest number of flagged antennas, which largely contribute to compromise the recovery of the mock halos. Our simulations also suggest that losses for JVLA are high at D ≥ 10.5′. In summary, for uGMRT and JVLA data, the inspection of the uv coverage and classical derivation of upper limits through injection are recommended rather than the usage of scaling relations similar to ours in Eq. (5).

Fig. 14 Recovered net flux density as a function of the injected flux density for various angular diameters and instruments. Red, magenta, blue, and green curves are injections in LOFAR (HBA at 144 MHz), uGMRT (band 3 at 400 MHz, 6 and 10 h on source are shown with solid and dotted lines, respectively), uGMRT (band 4 at 700 MHz), and JVLA (DnC+BnC array, L band at 1.5 GHz) datasets, respectively. The expected flux density integrated up to 3re of a representative radio halo with M500 = 5 × 1014 M⊙ is indicated by a dashed vertical line and can be exploited to estimate effective losses (see Table 3 and discussion in Sect. 6).

Fig. 15 Inner (≤200λ) uv coverage of the uGMRT observations used for injections: 6 h in band 3 (Dec ~ – 13°, top panel), 10 h in band 3 (Dec ~ +64°, middle panel), 9 h in band 4 (Dec ~ – 29°, lower panel).

7 Summary and conclusions

LoTSS-DR2 includes 140 non-detections of radio halos in Planck clusters (Botteon et al. 2022). We exploited these data to test the instrumental capabilities of LOFAR to recover diffuse extended emission and determine upper limits to the radio power of a possible halo.

Through the injection of mock visibilities simulating radio halos into the observed datasets, we estimated the flux density losses due to insufficient short baselines. We find that they are negligible (≲5–10%) for sources of sizes up to D = 14′ and reach fractions of ~20% at D = 18′. As common for LOFAR HBA data, our simulations were limited to baselines >80λ as well, meaning that more flux density can be recovered by including shorter baselines. For the first time, we systematically carried out tests on a large sample of datasets of varying quality, demonstrating that LOFAR is one of the facilities with the densest uv coverage in its inner part. It is thus able to recover large-scale emission with lower flux density losses with respect to other instruments. Moreover, the low frequency range of LOFAR allows us to explore host mass regimes that could barely be probed by facilities operating at higher frequencies.

We showed that non-detections of diffuse emission can be considered approximately independent of the uv coverage of LOFAR in LoTSS observations. We therefore explored the parameters that determine the flux density of the upper limits and found a relation with the noise of the image and the number of beams within the injected mock halo (depending on the resolution of the image and angular size of the mock emission). Our relation can be safely adopted to infer the upper limit if the subtraction of the discrete sources close to the cluster centre does not leave contaminating artefacts, which are enhanced at low resolution. Otherwise, subtraction artefacts typically drive the level of the limit, thus making the injection procedure and visual inspection still necessary to provide more reliable limits.

After excluding objects lacking redshift information, with extended radio galaxies, and with severe subtraction artefacts, we obtained upper limits for 75 Planck clusters in LoTSS-DR2. Our limits will be exploited in forthcoming statistical analyses (Zhang et al. 2023; Cassano et al. 2023, Cuciti et al., in prep.) and compared to the detected radio halos in LoTSS-DR2 to provide information on the populations, origin, and evolution of these sources.

Acknowledgements

We thank the referee for useful comments and suggestions that have improved the presentation of the paper. LB acknowledges Kamlesh Rajpurohit for providing us the JVLA data. GB and RC acknowledge support from INAF mainstream project ‘Galaxy Clusters Science with LOFAR’, 1.05.01.86.05. A. Botteon and R.J.vW. acknowledge support from the VIDI research programme with project number 639.042.729, which is financed by the Netherlands Organisation for Scientific Research (NWO). V.C. acknowledges support from the Alexander von Humboldt Foundation. A. Bonafede and A. Botteon acknowledge support from ERC Stg DRANOEL no. 714245 and MIUR FARE grant ‘SMS’. M.B. acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC 2121 ‘Quantum Universe’ – 390833306. DNH acknowledges support from the ERC through the grant ERC-Stg DRANOEL no. 714245. LOFAR (van Haarlem et al. 2013) is the Low Frequency Array designed and constructed by ASTRON. It has observing, data processing, and data storage facilities in several countries, which are owned by various parties (each with their own funding sources), and that are collectively operated by the ILT foundation under a joint scientific policy. The ILT resources have benefited from the following recent major funding sources: CNRS-INSU, Observatoire de Paris and Université d’Orléans, France; BMBF, MIWF-NRW, MPG, Germany; Science Foundation Ireland (SFI), Department of Business, Enterprise and Innovation (DBEI), Ireland; NWO, The Netherlands; The Science and Technology Facilities Council, UK; Ministry of Science and Higher Education, Poland; The Istituto Nazionale di Astrofisica (INAF), Italy. This research made use of the Dutch national e-infrastructure with support of the SURF Cooperative (e-infra 180169) and the LOFAR e-infra group. The Jülich LOFAR Long Term Archive and the German LOFAR network are both coordinated and operated by the Jülich Supercomputing Centre (JSC), and computing resources on the supercomputer JUWELS at JSC were provided by the Gauss Centre for Supercomputing e.V. (grant CHTB00) through the John von Neumann Institute for Computing (NIC). This research made use of the University of Hertfordshire high-performance computing facility and the LOFAR-UK computing facility located at the University of Hertfordshire and supported by STFC [ST/P000096/1], and of the Italian LOFAR IT computing infrastructure supported and operated by INAF, and by the Physics Department of Turin University (under an agreement with Consorzio Interuniversitario per la Fisica Spaziale) at the C3S Supercomputing Centre, Italy (Taffoni et al. 2022). This research made use of APLpy, an open-source plotting package for Python (Robitaille & Bressert 2012), Astropy, a community-developed core Python package for Astronomy (Astropy Collaboration 2013, 2018), Matplotlib (Hunter 2007), Numpy (Harris et al. 2020), and TOPCAT (Taylor 2005).

Appendix A Injections in PSZ2 G098.62+51.76

As a representative example of the various injections that we obtained in different NDE clusters, we report some of the images of PSZ2 G098.62+51.76, which were used to produce and discuss the plots shown in Sect. 4.

Fig. A.1 Injection of mock halos with D = 3′ (corresponding to the diameter of the yellow circle). The 2σ contour level is reported, where σ is the noise of the pre-injection map. The upper limit is obtained with Sinj,tot = 5 mJy.

Fig. A.2 Injection of mock halos with D = 5′ (corresponding to the diameter of the yellow circle). The 2σ contour level is reported, where σ is the noise of the pre-injection map. The upper limit is obtained with Sinj,tot = 10 mJy.

Fig. A.3 Injection of mock halos with D = 7.5′ (corresponding to the diameter of the yellow circle). The 2σ contour level is reported, where σ is the noise of the pre-injection map. The upper limit is obtained with Sinj,tot = 20 mJy.

Fig. A.4 Injection of mock halos with D = 10.5′ (corresponding to the diameter of the yellow circle). The 2σ contour level is reported, where σ is the noise of the pre-injection map. The upper limit is obtained with Sinj,tot = 35 mJy.

Fig. A.5 Injection of mock halos with D = 14′ (corresponding to the diameter of the yellow circle). The 2σ contour level is reported, where σ is the noise of the pre-injection map. The upper limit is obtained with Sinj,tot = 55 mJy.

Fig. A.6 Injection of mock halos with D = 18′ (corresponding to the diameter of the yellow circle). The 2σ contour level is reported, where σ is the noise of the pre-injection map. The upper limit is obtained with Sinj,tot = 80 mJy.

Appendix B Upper limits

In Table B.1, we summarise the main properties of the host cluster and the injected parameters used to obtain the upper limits.

References

All Tables

All Figures

	Fig. 1 Example of injection process. The yellow circle is centred on (RAinj, Decinj) and has a radius of r = 3re. Left: model image at 144 MHz of the mock halo following the exponential surface brightness profile of Eq. (1). Middle: pre-injection image. Right: post-injection image.
In the text

	Fig. 2 Restored images before injections in centre (left panels) and periphery (right panels) of PSZ2 G120.08-44.41 before (upper panels) and after (lower panels) subtraction of discrete sources.
In the text

Fig. 3 Restored images after injections in centre (panels A–C) and periphery (panels D–F) of PSZ2 G120.08-44.41 with different schemes: injection in the original dataset (‘INJ.’, panels A and D), injection in the discrete source-subtracted dataset (‘SUB. & INJ.’, panels B and E), and injection in the original dataset and subsequent subtraction of the discrete sources (‘INJ. & SUB.’, panels C and F). The yellow circle (centred on RAinj, Decinj) has a radius of 3re and contains Sinj = 0.8Sinj,tot, where Sinj,tot = 100, 50, and 20 mJy, for panels in Cols. 1–3, respectively.

In the text

Fig. 4 Point-to-point comparison of surface brightness of mock halos in PSZ2 G120.08-44.41 (‘Sub. & Inj.’ scheme; central and peripheral injections are shown in the upper and lower panel, respectively). Values at 20 and 50 mJy are multiplied by a factor of 5 and 2, respectively, to match the injection at 100 mJy. The one-to-one line is plotted in black. Insets show the corresponding radial profiles, whose points are obtained as the average brightness of four 90°-wide sectors, whereas the shaded region represents their standard deviation. The point-to-point plots show that the lower the brightness and the injected flux density, the higher the scatter around the one-to-one line, indicating the progressive apparition of emission patches. Analogously, the radial profiles indicate that the scatter increases at larger radii and lower brightness.

In the text

	Fig. 5 Net azimuthally averaged surface brightness profiles of the images shown in Appendix A for PSZ2 G098.62+51.76. The dashed black lines represent the theoretical injected profiles. The sampled profiles (Sinj, tot is reported in the legend) are obtained by subtracting the pre-injection from the post-injection contribution. The grey vertical line indicates r = 3re.
In the text

	Fig. 6 Fraction of recovered net flux density as a function of the injected flux density (within 3re) at various angular scales. Triangular markers indicate the flux density threshold at which the mock emission is not visible based on images in Appendix A. Non-negligible flux density losses (>15%) are expected for sources with angular sizes D > 15′ only.
In the text

Fig. 7 Azimuthally averaged profiles of images shown in Appendix A for PSZ2 G098.62+51.76. The dashed black lines represent the theoretical injected profiles. The sampled pre-injection and post-injection profiles are shown with black and coloured dots (see the total injected flux density in the legend), respectively. The grey vertical line indicates r = 3re. The grey horizontal line indicates the global 1σ noise level.

In the text

	Fig. 8 Logarithmic plot of the ratio of upper limit flux density against the noise of the map (SUL/σ) as a function of the number of beams (Nbeam) within the injected halo. The blue, red, and green lines are the fitted linear regressions (see Table 2) obtained with no cut in Nbeam, with Nbeam > 50 and with Nbeam < 50, respectively. The grey vertical lines are drawn at Nbeam = 30 and Nbeam = 50.
In the text

	Fig. 9 Distributions of redshifts and angular diameters of the 102 NDE galaxy clusters in our final sample. Top: distribution of redshifts. Bottom: distribution of the injected angular diameters under the assumption of re,inj = 200 kpc.
In the text

Fig. 10 Representative examples of NDE clusters showing artefacts introduced by the subtraction of discrete sources near the cluster centre. The radio contours are drawn at ±2σ and spaced by factors of 2. In all the panels, the dashed cyan circle indicates the cluster centre and has a fixed diameter of 2′. Left: NDE cluster excluded from our analysis due to severe (positive and/or negative) artefacts. Middle: NDE cluster with SQ = 1 due to the absence of artefacts. Right: NDE cluster with SQ = 2 due to the presence of moderate residuals.

In the text

	Fig. 11 Examples of injections for SQ = 2 cluster. Contours are drawn at 2σ of the pre-injection (left) image, and the cyan circle has a radius of 3re,inj. Two cycles of injections are performed with SUL derived from Eq. (5) and 1.5 × SUL. The upper limit is obtained with the second cycle, which leaves extended excess ~2 times brighter with respect to the pre-injection image.
In the text

	Fig. 12 Distribution of the ratio between the effective upper limit and the expected upper limit inferred from Eq. (5) for the SQ = 2 clusters.
In the text

Fig. 13 Distributions of flux density of confirmed and candidate radio halos (in blue) with 100 ≤ re ≤ 400 kpc from Botteon et al. (2022), our upper limits (in red), and upper limits from Cuciti et al. (2021a; in black), and George et al. (2021a; in yellow). Upper limits from the literature are re-scaled to 150 MHz by assuming α = 1.3. The samples are split in redshift bins as z < 0.3 (top panel) and z > 0.3 (lower panel).

In the text

Fig. 14 Recovered net flux density as a function of the injected flux density for various angular diameters and instruments. Red, magenta, blue, and green curves are injections in LOFAR (HBA at 144 MHz), uGMRT (band 3 at 400 MHz, 6 and 10 h on source are shown with solid and dotted lines, respectively), uGMRT (band 4 at 700 MHz), and JVLA (DnC+BnC array, L band at 1.5 GHz) datasets, respectively. The expected flux density integrated up to 3re of a representative radio halo with M500 = 5 × 1014 M⊙ is indicated by a dashed vertical line and can be exploited to estimate effective losses (see Table 3 and discussion in Sect. 6).

In the text

	Fig. 15 Inner (≤200λ) uv coverage of the uGMRT observations used for injections: 6 h in band 3 (Dec ~ – 13°, top panel), 10 h in band 3 (Dec ~ +64°, middle panel), 9 h in band 4 (Dec ~ – 29°, lower panel).
In the text

	Fig. A.1 Injection of mock halos with D = 3′ (corresponding to the diameter of the yellow circle). The 2σ contour level is reported, where σ is the noise of the pre-injection map. The upper limit is obtained with Sinj,tot = 5 mJy.
In the text

	Fig. A.2 Injection of mock halos with D = 5′ (corresponding to the diameter of the yellow circle). The 2σ contour level is reported, where σ is the noise of the pre-injection map. The upper limit is obtained with Sinj,tot = 10 mJy.
In the text

	Fig. A.3 Injection of mock halos with D = 7.5′ (corresponding to the diameter of the yellow circle). The 2σ contour level is reported, where σ is the noise of the pre-injection map. The upper limit is obtained with Sinj,tot = 20 mJy.
In the text

	Fig. A.4 Injection of mock halos with D = 10.5′ (corresponding to the diameter of the yellow circle). The 2σ contour level is reported, where σ is the noise of the pre-injection map. The upper limit is obtained with Sinj,tot = 35 mJy.
In the text

	Fig. A.5 Injection of mock halos with D = 14′ (corresponding to the diameter of the yellow circle). The 2σ contour level is reported, where σ is the noise of the pre-injection map. The upper limit is obtained with Sinj,tot = 55 mJy.
In the text

	Fig. A.6 Injection of mock halos with D = 18′ (corresponding to the diameter of the yellow circle). The 2σ contour level is reported, where σ is the noise of the pre-injection map. The upper limit is obtained with Sinj,tot = 80 mJy.
In the text