Revisiting the alignment of radio galaxies in the ELAIS-N1 field

Aims. Previous studies reported an alignment of the major axes of radio galaxies on various angular scales. Here, we study the alignment of radio galaxies in the ELAIS-N1 Low Frequency ARray (LOFAR) deep field, which covers an area of 25 $\rm deg^2$. \newline Methods. The low noise level of about 20$ \rm ~ \mu Jy/beam$ of the LOFAR deep field observations at 150 MHz enabled the identification of 447 extended ($>30 \rm ''$) radio galaxies for which we have measured the major axis position angle. We found that 95\% of these sources have either photometric or spectroscopic redshifts, which we then used for a three-dimensional analysis. \newline Results. We show the distribution of the position angles of radio galaxies in the ELAIS-N1 field and perform multiple statistical tests to check whether the radio galaxies are randomly oriented. We found that the distribution of position angles is consistent with being uniform. Two peaks around position angles of 50 and 140$\rm~ deg$ are spurious and are not caused by an alignment, as shown by a 3D analysis. In conclusion, our results do not support a 2D or 3D alignment of radio galaxies on scales smaller than $\sim 4 \rm ~ deg$.


Introduction
The cosmological principle is an assumption in modern cosmology which states that the Universe is (statistically) isotropic and homogeneous on suitably large scales ( 100 Mpc).Multiple observations have investigated the degree of anisotropy in the cosmic microwave background (Bennett et al. 1996;Hansen et al. 2004;Planck Collaboration et al. 2016, 2020) confirming the principle of homogeneity and isotropy of the Universe.However, several authors have reported an intriguing alignment of the linear polarisation of quasars (Stockman et al. 1979;Hutsemekers 1998;Hutsemékers & Lamy 2001;Jain et al. 2004;Cabanac et al. 2005;Pelgrims & Cudell 2014;Slagter & Miedema 2021;Friday et al. 2022).Interestingly, they found an alignment mainly occurring in groups of 10-30 objects and on potentially Gpc scales.Some other studies focused on the alignment of radio galaxy jets (e.g., Sanders 1984;Kapahi et al. 1985;West 1991;Joshi et al. 2007;Tiwari & Jain 2013), some of which support a possible departure from the cosmological principle.Taylor & Jagannathan (2016) studied the spatial distributions of the major axis position angle of radio galaxies in the ELAIS-N1 Giant Metrewave Radio Telescope (GMRT, Ananthakrishnan 1995) deep field.They claimed the existence of a 2D alignment around PA ∼ 140 • over an area of ∼ 1.7 deg 2 .However, for lack of the redshifts of the host galaxies, they did not perform a 3D analysis.The first attempts to detect an alignment on larger scales were made by The full Table A.1 is only available in electronic form at the CDS via anonymous ftp to cdsarc.cds.unistra.fr(130.79.128.5) or via https://cdsarc.cds.unistra.fr/cgi-bin/qcat?J/A+A/ Contigiani et al. (2017) and Panwar et al. (2020) who used catalogue data from the Faint Images of the Radio Sky at Twenty-cm (FIRST, Becker et al. 1995;Helfand et al. 2015) and the TIFR GMRT Sky Survey (TGSS, Intema et al. 2017).They detected a signal over a scale smaller than 2 • , but did not find strong evidence for a 3D alignment.For the first time, Blinov et al. (2020) explore the alignment of parsec-scale jets finding that their radio sources do not show any global alignment.However, Mandarakas et al. (2021), with a similar but larger sample, detected a strong signal of an alignment of parsec-scale jets in multiple regions of the sky.Nevertheless, the redshift distribution of their sources spans a wide range, 0 < z 1.5 Most recently, Osinga et al. (2020) searched for alignment using 7555 extended sources from the first data release of the Low Frequency ARray Two metre Sky Survey (LoTSS, Shimwell et al. 2019).However, despite their use of host redshifts, they could only detect a 2D alignment of the position angles of the radio galaxies over a scale of 5 • and could not exclude the possibility that the signal arises from systematic effects.Although multiple studies have now presented evidence for a 2D or 3D alignment, an explanation for such a phenomenon is lacking.West (1991), Hutsemékers et al. (2014) Pelgrims & Hutsemékers (2016) found an alignment between the radio and optical emissions from active galactic nuclei (AGN) and the surrounding large-scale structure.Moreover, Malarecki et al. (2013Malarecki et al. ( , 2015) ) showed that giant radio galaxies (Willis et al. 1974) have a tendency to grow in a direction perpendicular to the major axes of galaxy overdensities.However, the connection between the orientation of radio galaxy jets and the large-scale structure is unclear.
In this paper, we revisit the alignment of radio galaxies jets in

Methods
We inspected the ELAIS-N1 LOw-Frequency ARray (LOFAR, van Haarlem et al. 2013) deep field (Sabater et al. 2021).With an effective observing time of 163.7 h, it reaches a root mean square noise level at 150 MHz lower than 30 µJy beam −1 across the inner 10 deg 2 and below 20 µJy beam −1 in the very centre.The ELAIS-N1 LOFAR Deep Field (ELDF) is centred on 16h11m00s + 55 • 00 00 (J2000) and it covers an area of about 25 deg 2 .The 6" resolution of the radio image ensures a robust classification of the sources and, most importantly, the identification of the hosts and radio features such as jets and hotspots.

The sample of extended radio galaxies
We searched for all the ERGs with a largest angular size (LAS) larger than ∼ 30" within an area of ∼ 25 deg 2 .We measured the LAS as the distance between the opposite ends of the ERGs.However, this method can overestimate the size of the Fanaroff-Riley type II (FRII, Fanaroff & Riley 1974) as commented in Kuźmicz & Jamrozy (2021).Thus, for such ERGs, we measured the LAS as the distance between the two hotspots, whenever identified on the VLA Sky Survey images (Lacy et al. 2020).The radio position angles (RPAs) were manually measured (by using Aladin * , Bonnarel et al. 2000) in the range [0,180) degrees as the angle between the source's major axis and the local meridian, from N through E. For straight (or only slightly bent) FRI and FRII, the RPA is either that of the inner jets (FR I) or that of the direction connecting the two hotspots (FR II).In the case of bent sources (e.g., Wide-Angle-Tailed RGs), measuring the RPA is less trivial.For such cases, we measured the RPA in the vicinity of the core where usually the jets are not bent yet and flagged them as uncertain measurements.We carefully avoided measuring the RPA of overlapping sources unless the morphology of the ERGs was very clear.A large number of optical and infrared surveys, such as the Wide-Field Infrared Survey Explorer (WISE, Cutri & et al. 2012;Cutri et al. 2013;Schlafly et al. 2019;Marocco et al. 2021), the Sloan Digital Sky Survey (SDSS, York et al. 2000), the Legacy survey (Dey et al. 2019) and the Panoramic Survey Telescope and Rapid Response System (Pan-STARRS, Flewelling et al. 2020) enabled the identification of the host galaxies (see Kondapally et al. 2021;Andernach et al. 2021;Simonte et al. 2022, for further details on the host identification and radio source classification).We looked for available redshifts (either spectroscopic or photometric) in multiple catalogues such as Rowan-Robinson et al. (2013), Bilicki et al. (2014), Bilicki et al. (2016), Beck et al. (2016), Duncan et al. (2021), Beck et al. (2021), Zhou et al. (2021), Wen & Han (2021) and Duncan (2022).If for a single source multiple photometric redshifts were available, we computed their mean and error by taking the standard deviation of  (2021) the faintest galaxies have a maximum redshift of around 1.3.Thus, we assumed a redshift in the range of 1.1-1.5 for those host galaxies without redshift listed in the literature.This assumption will not affect our analysis as we will use only those sources with a redshift reported in the literature for the 3D analysis.We found 447 ERGs for which we provide redshift, LAS, largest linear size (LLS) and RPA.We show some of our ERGs in Table A.1 and the full list will be made available at the CDS and through the VizieR service † (Ochsenbein et al. 2000).To test the alignment in the region inspected by Taylor & Jagannathan (2016), we located all sources these authors had used (their Fig. 2) and measured their RPAs.Some of these RGs have an angular size smaller than 30 .The resolution of 6" of the LO-FAR images does not enable reliable measurement of the RPA of the smallest sources and we flagged these measurements as uncertain.We had to discard 9 RGs used by Taylor & Jagannathan (2016) as 8 of them are separate sources and one is a spiral galaxy (see Appendix A).However, we were able to add 24 more ERGs within the sky area they studied that we were able to identify using the LOFAR data.In Tab. 1 we compare our sample with previous lists of RGs used for the RPA analysis.In this work, we analysed a field ∼ 10 times larger than that of Taylor & Jagannathan (2016), but much smaller than those used by Contigiani et al. ( 2017), Panwar et al. (2020) and Osinga et al. (2020).Nevertheless, our sample has the largest RGs sky density in the central region (241.5 which is reported in the last row, while the second-last row shows the RGs sky density considering the full ELDF. use five different tests for (non-)uniformity of angles: 1.The Kolmogorov-Smirnov (KS) test compares the underlying distribution of the sample of the RPA against a given distribution, which in our case is a uniform distribution.The null hypothesis is that the two distributions are identical and the closer the p-value is to zero the more confident we are in rejecting the null hypothesis.A common threshold used to reject the null hypothesis of the two distributions being drawn from the same population is a p-value p<0.05, which means that there is only a 5% chance that the two samples are in fact drawn from the same population.
2. Pearson's χ 2 test for uniformity tests the null hypothesis stating that the frequency distribution of certain events observed in a sample is consistent with a particular theoretical distribution (in our case a uniform one).As with the KS test, the smaller the pvalue the more likely it is that the two distributions are different.This test is performed with binned data and, in our case, we used 18 bins which are 10 • wide.3. Our set of RPAs belongs to the category of circular data (Fisher 1993) which are fundamentally different from linear data due to their periodic nature.The Rayleigh test (Mardia & Jupp 2000) assesses the uniformity of circular data.To this end, this test compares the test statistic of the unit vector, resulting from the sum of all the vectors pointing towards the different angles of the sample, with the same statistics estimated from a uniformly distributed sample.The null hypothesis of such test is that the data are uniformly distributed over the circle.The test statistic is the mean resultant length of the unit vector and it is defined as where n is the size of the sample and the angles θ i are the RPAs multiplied by two since these are orientations (axial vectors) in the range [0 • , 180 • ) while the Rayleigh test is performed on the range [0 • , 360 • ).R can range from 0 to 1.This statistic is zero for a uniform distribution, thus it is reasonable to reject uniformity when R is large.It is worth mentioning that this test is not sensitive to non-uniform distributions that have R = 0.An example is a bimodal distribution with two peaks that are 180 • apart as every vector pointing towards a certain direction is cancelled by a vector pointing along the opposite direction.This issue can mildly affect our analysis since the major peaks in our distributions of the RPAs are 180 • apart once the RPAs are multiplied by two (see Sec. 3 below).4. The semi-variance (Cressie 1993) is a statistical tool used in spatial analysis to measure the dispersion of a certain variable on different scales.It is defined as follows: where m(d) is the number of pairs separated by an (angular) distance in the range [d, d + δd] (we used δd = 0.2 • ) and s is the variable measured at the vector location x i and in our case is the RPA of the ERGs.The semi-variance is constant over all angular scales when the distribution of the variable s is uniform.A value for the semi-variance smaller than what is predicted by a uniform distribution at a certain scale indicates an alignment of the ERGs.On the other hand, a larger semi-variance suggests a larger dispersion than expected from a random distribution, indicating that no alignment is present on that scale.We performed a simple Monte-Carlo simulation to infer the value of the semi-variance of randomly distributed ERGs on different angular scales.We generated 447 (which is the size of our sample) random angles uniformly distributed in the range [0, 180) which have the same spatial distribution of the ERGs in our sample and we computed the semi-variance on different angular scales.We repeated the operation 10000 times and then averaged the semi-variance values on the different scales.We folded the data in circularity to take into account the periodicity of the RPAs.On every scale, we obtained a constant semi-variance of 0.82, consistent with the result from Taylor & Jagannathan (2016).The error on the semi-variance, σ SM was estimated by calculating the standard deviation of the 10000 values on each angular scale.5. Finally, we probed the alignment of the ERGs at different angular scales using the dispersion measure analysis (Jain et al. 2004).The dispersion measure is defined as the inner product between a certain position angle θ and the RPAs, θ k , of the n closest sources to a certain i-th ERG (including the source itself) and it is an indication of the alignment of the ERGs.Following Jain et al. ( 2004), Contigiani et al. (2017) and Osinga et al. (2020), it can be shown that the maximum dispersion measure around the source i is The closer D i,n | max is to 1, the more aligned the n galaxies are.
The statistic, S n used to test the (non-)uniformity of the distribution of the RPAs is the average of the D i,n | max calculated for each source of the sample.This statistic computed from our dataset is compared to the same statistics coming from Monte-Carlo simulated samples, S n,MC .To compute S n,MC we generated 447 randomly oriented ERGs with the same spatial distribution of our sources and followed the formalism described in Jain et al. (2004), Contigiani et al. (2017) and Osinga et al. (2020).We repeated the calculation of S n,MC 10000 times and estimate the average, S n,MC , and the error, σ n,MC , as the standard deviation of 10000 generated statistics.The significance level for rejecting the null hypothesis that a sample of ERGs is randomly oriented is found through a one-tailed significance test, expressed as: where Φ is the cumulative normal distribution function.The closer the significance level is to 0 the more confident we are in rejecting the hypothesis of uniformity.Since the number of nearest neighbours can be translated to an angular scale extending to the n-th nearest neighbour, we can probe multiple angular scales varying n.To do so, we calculated the maximum angular distance between the relevant ERG and the n-th closest neighbour and took the median value among the 447 sources.The same analysis can be implemented considering the 3D position of the ERGs to test whether a 3D alignment, i.e. between sources that are physically close to each other, is present.We approximated the redshift of the source with the average redshift estimated for each ERGs without taking into account the error and we did not include those sources without a redshift value reported in the literature.The uncertainties of some redshift estimations might mildly affect the analysis: in fact, while ERGs with z < 1 have a redshift error of about 0.05, for more distant sources, which represent 30% of our sample, the error increases to 0.2.We then converted the redshift to comoving distance and measured the 3D comoving distance between all the ERGs in our sample.Moreover, Jain et al. (2004) verified that the variance of the statistic S n is inversely proportional to the size of the sample which means that, compared to Contigiani et al. (2017) and Osinga et al. (2020) who used much larger samples, we are more affected by the shot noise.

Results
In this section, we present the distribution of the RPAs in the ELAIS-N1 field.We initially focus on the inner region studied by Taylor & Jagannathan (2016) and then expand the analysis to the entire ELDF.

Alignment in the central part of ELAIS-N1
Here, we look at the distribution of the RPAs in the inner ∼1.7 deg 2 of the ELAIS-N1 field (241.5 < RA < 243.75, 53.9 < DEC < 55.2), where Taylor & Jagannathan (2016) found a statistically significant alignment of radio galaxies.We recall that 9 radio sources they used in their analysis are not actual radio galaxies and we could add 24 more ERGs.Thus, the sample for such analysis consists of 78 ERGs, of which 19 are flagged as uncertain RPA measurement.We show the distribution of the RPAs in the inner region of the ELAIS-N1 field in Fig. 1.The blue histogram shows the distribution of the total sample of RPA in this region, while in the red histogram the uncertain measurements are excluded.The figure clearly shows a peak at RPAs around 140 • in agreement with Taylor & Jagannathan (2016).We then carried out the statistical tests explained in Sec.2.2 and found a p-value of 0.66 and 0.31 for the KS test and the χ 2 test, respectively.The latter test is valid for large samples and it is customary to recommend, in applications of the test, that the smallest expected number in any bin should be 5 (Cochran 1952).We performed the test using 13 bins with a width of 15 • which lead to an expected value of about 6.5 elements per bin.The resulting p-value, in this case, is 0.23.Concerning the Rayleigh test, we found a mean resultant length R = 0.009 which results in a p-value=0.96.Thus, even though the distribution shows a clear peak, we cannot reject the hypothesis of uniformity of the RPAs in this region.Moreover, the analysis involving the semivariance (Fig. 2) shows that there is no correlation between the RPAs of the ERGs, located at different positions of the sky, at any angular scale.Here, the blue line and points are the values estimated by using randomly generated data which have the same spatial distribution of the 78 ERGs in the inner region of the ELAIS-N1 field, while the orange points are the result of the analysis performed on our dataset.We did not perform an analysis based on the dispersion measure (that is the 5th method listed in Sec.2.2) due to the smaller number of ERGs when restricting the study to the inner region of the field.As a matter of fact, with a sample of only 78 objects, we are certainly dominated by the shot noise (Jain et al. 2004) which would cancel out any signal unless the alignment is very strong, which does not seem to be the case here.
We performed the statistical tests on the sample of 59 ERGs for which we could measure a reliable RPA as well.We obtained a p-value of 0.10, 0.01 and 0.46 for the KS, χ 2 and Rayleigh test, respectively.The result of the χ 2 test holds when considering bins with a width of 15 • .Nevertheless, this is the only test which suggests an alignment of the ERGs in the inner region as also the semi-variance test applied to this smaller sample cannot reject the hypothesis of a uniform distribution.
The sensitivity of the LOFAR (20 µJy/beam) and GMRT (10 In order to attempt to reproduce the Taylor & Jagannathan ( 2016) results, we extracted the positions, sizes and RPAs of the RGs from their Fig. 2 as follows: the end points of all vectors were digitized with the g3data software, and saved as RA, DEC in degrees.Then, we reviewed the RPA measurements and could closely match the histogram shown in their Fig. 3.We ran our first four statistical tests on the recovered data, but found that none of them is able to reject the hypothesis of uniformity.In particular, for the Rayleigh test, we obtained a mean resultant length of 0.09 from our analysis of these data, which is highly discrepant from the value of 0.68 derived by Taylor & Jagannathan ( 2016) that led them to conclude non-uniformity of RPAs.The origin of this difference is uncertain, although we note that if we omit to multiply the RPAs by a factor of two (a step which is required, since the test assesses uniformity over a circle and the RPAs are distributed over [0, 180)) then we obtain an erroneous mean resultant length of 0.64, which is much closer to the value quoted by Taylor & Jagannathan (2016).

Alignment in the entire ELAIS-N1 field
We show the distribution of the RPAs of the radio galaxies in the ELDF in Fig. 3.The blue histogram represents the total sample, while the red histogram shows the distribution for the 377 certain sources, i.e. those ERGs that do not show a complex morphology and for which we could accurately measure the RPA.The black line denotes the expected number of objects per bin if the distribution were uniform.Now, we performed the same statistical tests considering the total sample.The results, with a p-value equal to 0.71, 0.33 and 0.88 for the KS test, χ 2 test and Rayleigh test respectively, suggest that the uniformity holds when including the entire field as well.These results are also confirmed by the analysis of the semi-variance.We measured the semi-variance in our sample, shown by the orange points in Fig. 4. The blue line and points are the semi-variance values estimated from randomly generated data and the shadowed region represents the 2σ SM values.The larger uncertainties on the largest scale are due to poor statistics since not many pairs are separated by such large distances.Overall, there is no clear evidence for a convincing signal in favour of an alignment as the orange points are always consistent with 0.82 within the error.
Finally, we show the results of the 2D (black line) and 3D (blue line) dispersion measure tests in Fig. 5.The significance level, S L, is plotted as a function of the number of nearest neighbours, n, and angular scale in degrees.Following previous studies (e.g., Contigiani et al. 2017), a commonly used criterion for the presence of an alignment signal is S L <0.03 (log(SL) < −1.5).As mentioned in Sec.2.2, we are more affected by the shot noise due to the comparatively smaller size of our sample.However, a minimum significance level of about 0.2 in Fig. 5 suggests there is no evident signal, neither in the 2D nor in the 3D analysis, at any scale.These results also hold when considering only the ERGs with reliable RPA measurement.Even though the tests suggest that radio galaxies are randomly & Jagannathan 2016 as well).The Poisson distribution gives the probability that a given number of observations fall within an interval of values knowing the average frequency of that particular event.Thus, using such a distribution we find that the two peaks are ∼ 2.5σ (for RPAs between 50 • − 60 • ) and ∼ 1.5σ (for RPA between 140 • − 150 • ) above the average.In Fig. 6 we show the spatial and redshift distributions of the ERGs with an orientation between 50 • -60 • (upper panel) and 140 • -150 • (lower panel).We selected the ERGs up to redshift 1.5 since the majority of ERGs at larger redshifts either do not have a redshift estimate in the literature or have very large errors.The black rectangles highlight the region inspected by Taylor & Jagannathan (2016).In both cases, there is no 3D alignment of ERGs as the redshifts span a range from 0.1 z 1.5.

Discussion and summary
The tidal torque theory predicts that the angular momentum of the dark matter proto-halos is acquired during their formation which occurs along the entire evolution of the large-scale structure of the universe (Peebles 1969;Doroshkevich 1970;White 1984;Porciani et al. 2002;Schäfer 2009).As a result, an alignment between optical galaxies and the large-scale structure (e.g.filaments and sheets) is expected (Hu et al. 2006;Joachimi et al. 2015;Kirk et al. 2015).In a first attempt to study this alignment Hawley & Peebles (1975) found a small departure from isotropy in the distribution of the orientation angle, measured as the angle between the major axis of the galaxy and the local meridian.Lee (2004) argued that the observed large-scale coherence in the orientation of nearby spiral galaxies found by Navarro et al. (2004) can be fully explained by the tidal torque theory.Others have tried to look at a possible alignment of galaxies and most of these found that the minor axes of early-type galaxies are preferentially oriented perpendicular to the host filament (Tempel et al. 2013;Tempel & Libeskind 2013;Hirv et al. 2017), while late-type galaxies have spin axes parallel to the closest filament (Tempel et al. 2013;Tempel & Libeskind 2013;Hirv et al. 2017;Blue Bird et al. 2020;Kraljic et al. 2021;Tudorache et al. 2022).However, some conflicting results have been found (Jones et al. 2010;Zhang et al. 2015;Pahwa et al. 2016;Krolewski et al. 2019).Recently, Rodriguez et al. (2022), by using the IllustrisTNG simulations (Nelson et al. 2019), found an alignment with the large-scale structure of red galaxies in the centres of galaxy clusters and groups.They then speculated that this anisotropy in the orientation of the central galaxies is the consequence of a concatenation of alignments.Starting from the alignment between the central galaxy and the host cluster (Yuan & Wen 2022), eventually, the host halo aligns with the structures surrounding it.Some work found that there is a mild preference for radio jets to align with the minor axis of the galaxy host (Kotanyi & Ekers 1979;Battye & Browne 2009;Kamali et al. 2019;Vazquez Najar & Andernach 2019).Assuming that the alignment between radio jets and optical galaxies is real, one could in principle look at the alignment between the radio galaxies and the large-scale structure (e.g., West 1991).Nevertheless, some opposing results regarding the orientations of radio jets have been found (Schmitt et al. 2002;Verdoes Kleijn & de Zeeuw 2005;Hopkins et al. 2012) casting doubts on this assumption.In this work, we revisited the alignment of radio jets in the ELAIS-N1 field.We inspected the LOFAR ELAIS-N1 deep field in which we identified the host galaxies of 447 ERGs whose radio emission extends over at least ∼ 0.5 .We measured the RPA of the major radio axis (assuming it is a tracer of the underlying radio jets direction) and studied their distribution by using a number of statistical tests, none of which is able to reject the null hypothesis of uniform orientations.Similar results are obtained when restricting the analysis to the region inspected by Taylor & Jagannathan (2016).Only when restricting the sample to the 59 ERGs with reliable RPA measurement in the inner region, the χ 2 test returns a p-value=0.01(i.e. it attributes a 1% chance of the result being a statistical fluctuation).However, none of the other statistical tests on this sample is able to reject the hypothesis of uniformity of the RPA distribution.We recovered the data used by Taylor & Jagannathan (2016) for their analysis and showed that, even with such sample, we could not obtain the same results.Furthermore, we found that the redshifts of ERGs with orientations near the two peaks (around 50 • and 140 • ) span a wide range, 0.1 z 1.5, strongly arguing against the idea of a 3D alignment of radio galaxies.Other reports of a 3D alignment (e.g., Contigiani et al. 2017;Panwar et al. 2020) have not been statistically significant.However, several studies reported a 2D alignment (Contigiani et al. 2017;Panwar et al. 2020;Mandarakas et al. 2021) over angular scales similar to those that we studied.The maximum angular scale we could explore is ∼ 4 • (see Fig. 4) which is the scale over which Osinga et al. (2020) found a 2D alignment.The combination of the two results might suggest that the 2D alignment of radio galaxies may exist on scales larger than those probed by our analysis.

Fig. 1 .
Fig. 1.Distribution of the RPAs of the 78 ERGs (blue histogram) that we found in the inner region of the ELDF and of the 59 certain sources (red histogram).The black line shows the expected number of objects per bin for a uniform distribution of 78 ERGs.

Fig. 2 .
Fig. 2. Estimate of the semi-variance on different angular scales in the inner region of the ELAIS-N1 field.The blue line and points are the semi-variance values obtained for randomly generated position angles with the same spatial distribution of the 78 ERGs.The shadowed region represents the 2σ SM values.The orange points are estimated from our dataset.

Fig. 3 .
Fig. 3. Distribution of the RPAs of the 447 ERGs we found in the ELDF (red histograms) and of the 377 certain sources (red histogram).The black line shows the expected number of objects per bin for a uniform distribution considering the total sample.

Fig. 4 .
Fig. 4. Estimate of the semi-variance on different angular scales.The blue line and points highlight the constant value of the semi-variance for randomly generated position angles with the same spatial distribution of the 447 ERGs in our sample.The shadowed region represents the 2σ SM values.The orange points are the semi-variance values of our sample.

Fig. 5 .
Fig. 5. Significance level of the dispersion measure test (SL) as a function of the nearest neighbours (n, lower abscissa) and angular scale in degrees (upper abscissa).The black line shows the results of the 2D analysis while the 3D analysis is shown with the blue line.Such a large SL (>> 0.03) shows that no alignment is present in the ELAIS-N1 field at any scale in our analysis.

Fig. 6 .
Fig. 6.Spatial distribution of 39 ERGs with RPA between 50 • and 60 • (upper panel) and 31 ERGs with RPA in the range 140 • − 150 • .The colorbar shows the redshifts of the ERGs.The analysis was restricted to z ≤ 1.5.The black box represents the field of Taylor & Jagannathan (2016).