Compact Symmetric Objects. II. Confirmation of a Distinct Population of High-luminosity Jetted Active Galaxies

Compact symmetric objects (CSOs) are compact (<1 kpc), jetted active galactic nuclei (AGN), whose jet axes are not aligned close to the line of sight, and whose observed emission is not predominantly relativistically boosted toward us. Two classes of CSOs have previously been identified: approximately one-fifth are edge dimmed and the rest are edge brightened. We designate these as CSO 1s and 2s, respectively. This paper focuses almost exclusively on CSO 2s. Using complete samples of CSO 2s we present three independent lines of evidence, based on their relative numbers, redshift distributions, and size distributions, which show conclusively that the vast majority (>99%) of CSO 2s do not evolve into larger-scale radio sources. These CSO 2s belong to a distinct population of jetted AGN, which should be characterized as “short-lived,” as opposed to “young,” compared to the classes of larger jetted AGN. We show that there is a sharp upper cutoff in the CSO 2 size distribution at ≈500 pc. The distinct differences between most CSO 2s and other jetted AGN provides a crucial new time domain window on the formation and evolution of relativistic jets in AGN and the supermassive black holes that drive them.


INTRODUCTION
The first indication of relativistic motion in the jets of active galaxies was the asymmetric large-scale jet in M87 discovered by Curtis (1918).The next was arguably the discovery of rapid flux density variations in blazars (Dent 1965a,b), which were quickly shown by Rees (1966Rees ( , 1967) ) to be due to relativistic motion of the emission regions towards the observer.In spite of this development, observations of the synchrotron selfabsorption cutoff frequencies of radio sources with flat spectra led to the hypothesis that an "inverse Comp-ton catastrophe" imposes an upper limit of ∼ 10 12 K on the brightness temperatures of compact radio sources (Kellermann & Pauliny-Toth 1969).This appeared, at first, to be supported by Very Long Baseline Interferometry (VLBI) observations, but in these calculations the possibility of relativistic bulk motion towards the observer (Rees 1966(Rees , 1967) ) was not taken into account.
The first phase-coherent astronomical image ever obtained in any energy band, including the optical band, having a resolution significantly less than one arc second was produced in the first "hybrid map", which showed an asymmetric one-sided radio jet (Wilkinson et al. 1977).Such core-jet structures were soon shown to predominate in compact radio sources (Readhead et al. 1978;Readhead 1980), making it clear that relativistic beaming determines the apparent morphology and the observed brightness temperatures of most compact radio sources at cm wavelengths.Nevertheless, the inverse Compton catastrophe hypothesis continued to propagate, but, as shown by Readhead (1994), when relativistic beaming is taken into account, the brightness temperatures drop to ∼ 10 11 K, and are consistent with equipartition between the magnetic field and particle energy densities in the emission regions.
It should therefore be clear that relativistic beaming greatly complicates the physical analysis of the observed radio emission of compact radio sources.In order to overcome such complications, which introduce large uncertainties in the physical properties, such as the magnetic field strength, the particle energy densities, the pressures, and the total energies of the emission regions, Wilkinson et al. (1994), hereafter W94, introduced the Compact Symmetric Object (CSO) classification of compact radio sources.Due to the morphological symmetry of the emission on either side of the nucleus, these objects are clearly not exhibiting strongly beamed emission towards the observer.
Unfortunately, a number of jetted-AGN have been misidentified as CSOs or CSO candidates in the literature, and many jetted-AGN whose axes are close to the line of sight, and whose observed emission is strongly beamed towards us, have crept into this class.This paper is the second of three on the morphological radio properties of CSOs in which we explore CSO phenomenology uncontaminated by objects that have been mis-identified as CSOs.In the first paper (Paper 1: Kiehlmann et al. in press) we added two new criteria, based on variability and speed, to the CSO selection criteria and undertook a detailed survey of the literature, which enabled us to identify 79 bona fide CSOs.From the 79 bona fide CSOs we determined the numbers in three complete samples1 from which, in this paper (Paper 2), we show that ≳ 99% of CSO 2s form a class of jetted-AGN that is both distinct from other jetted-AGN and exhibits a sharp cutoff in size at ≈ 500 pc, and a corresponding cutoff in age at ≈ 5000 yr, so that only fewer than 1% of CSO 2s might possibly go on to form the larger classes of jetted-AGN, such as Fanaroff and Riley Type I (FRI) and Type II (FR II) objects (Fanaroff & Riley 1974).In the third paper (Paper 3: Readhead et al. in press) we discuss the evolution of CSO 2s and show that while they are nearly all "short-lived" compared to the classes of larger jetted-AGN, only a minority of them are "young".Note that FR I and FR II objects have sizes in the ∼20 kpc -several Mpc range, and therefore clearly have ages much longer than the vast majority of CSO 2s.We should avoid the implicit assumptions involved in calling all CSO 2s "young", which obscure the true nature and importance of this class of jetted-AGN.It is critically important, therefore, to recognize the distinction between the terms "young" and "shortlived", which otherwise obfuscate the phenomenology of the CSO class.
In an important development in the study of CSOs, Tremblay et al. (2016) showed that there are two major morphological classes of CSOs: edge-dimmed objects, which we designate as CSO 1s, and edge-brightened objects, which we designated as CSO 2s.Paper 1 confirms their finding and this paper deals almost exclusively with CSO 2s.
As discussed in detail in Paper 1, in CSOs, two emission regions are seen straddling the center of activity, making it clear that these cannot be strongly relativistically boosted, otherwise the object would be seen as a one-sided asymmetric "core-jet" object as is the case in the vast majority of compact radio sources observed at cm wavelengths (Kellermann et al. 1998;Lister et al. 2019).
Individual CSO 2s undergo appreciable evolutionary structural changes on timescales of years that can therefore be studied without the complications of relativistic beaming.The bulk flows along their jets and their speeds of advance into the interstellar medium can be measured directly.We argue that CSO 2s provide a uniquely accessible time domain laboratory for the study of relativistic jets (Blandford et al. 2019) and the SMBH central engines that drive them, because they are shortlived compared to the classes of larger jetted-AGN, rather than young, and hence pass through all stages of their lives as CSO 2s, which are therefore available for detailed study in all phases of their lives.It is important to distinguish between CSOs that have small sizes because they are "short-lived" compared to larger classes of jetted-AGN, and CSOs that have been stalled by the interstellar medium of their host galaxies and therefore stopped growing in size.We propose the hypothesis that such stalled CSOs are likely to be edgedimmed and hence fall into the CSO 1 class.We also note in passing that this could be of great importance to feedback.As we show in Paper 3, the most luminous CSO 2s that are the subject of this study have not been stalled -their hot spots are separating on average at ∼ 0.4c, and their maximum lifetimes are ∼ 5000 yr.For the purposes of this study, although stalled CSO 2s are of great potential interest, we do not consider them further in these three papers.A minority of the less lu-minous CSO 2s in our study might possibly be stalled systems and should also be considered in that light.But this is beyond the scope of the present study.
By the early 1990s, three bona fide CSO 2s had been definitively identified in the complete sample of 65 radio sources studied by Pearson & Readhead (1988).Despite the small size of the CSO sample, and entirely because it was part of a complete sample, this sample of only three CSO 2s was enough to enable a number of the most critical questions about CSO 2s to be addressed by Readhead et al. (1994), hereafter R94, including their relationship to the larger jetted-AGN, their lifetimes, and their energy requirements.R94 concluded that CSO 2s form a distinct population of compact jetted-AGN, and that there must be a physical reason for this which provides a unique window on the central engines that drive AGN.R94 also suggested that CSO 2s might be the result of the capture of a single star by a SMBH in an otherwise quiescent elliptical galaxy nucleus.This possibility was also suggested more recently by An & Baan (2012).
All of these properties of CSO 2s were discussed in more detail, and confirmed, in Readhead et al. (1996), hereafter R96.Nevertheless, in spite of their distinction, CSO 2s have attracted comparatively little attention among jetted-AGN enthusiasts.We explore the characteristics of CSO 2s in considerably more detail in this paper and in Paper 3.
The CSOs are a subset of AGN, but by studying a restricted well-defined sample of CSOs we aim to understand them in depth and gain new insights into the physics and formation of jetted-AGN.Although it is not a primary goal of these papers, we discuss the relationship of CSOs to other classes of AGN where appropriate in this paper and in Paper 3. To place CSOs in the broader context of compact radio sources associated with AGN, the reader is referred to the comprehensive review of O'Dea & Saikia (2021a), hereafter OS21.
Throughout this paper we adopt the convention S ν ∝ ν α for spectral index α, and use the cosmological parameters Ω m = 0.27, Ω Λ = 0.73 and H 0 = 71 km s −1 Mpc −1 (Komatsu et al. 2009).We do this for consistency with our other papers.None of the conclusions would be changed were we to adopt the best model of the Planck Collaboration (Planck Collaboration et al. 2020).

COMPLETE SAMPLES OF JETTED-AGN
The disposition of the CSOs we consider in this study, amongst the CSO 1 and CSO 2 classes, the CSOs with spectroscopic redshifts and those without, and the CSOs In this paper we deal almost exclusively with the 17 CSOs in these complete samples that have spectroscopic redshifts.in complete samples is shown in Table 1.The classification of CSO 1s and CSO 2s that we use here is discussed in more detail in Paper 3. We use only complete samples for statistical tests in this paper.Other methods for making statistical tests, which are not based on complete samples, must introduce some assumptions regarding the population under study, and we wish to avoid making such assumptions.We use three complete samples extensively (see Table 2): (1) The 5 GHz Pearson-Readhead (PR) complete sample (Pearson & Readhead 1988) based on the MPIfR/NRAO S4 and S5 surveys (Pauliny-Toth et al. 1978;Kuehr et al. 1981); (2) The first Caltech-Jodrell (CJ1) 5 GHz complete sample (Polatidis et al. 1995;Xu et al. 1995)l; (3) The Peacock-Wall (PW) 2.7 GHz complete sample (Peacock & Wall 1981;Wall & Peacock 1985). 2here are 282 objects in the union of these three complete samples and these are listed in Table 7 in the Appendix.In our analysis in this paper, we exclude M82 (3C 231) which is in the PR and PW samples, but is a starburst galaxy and not a jetted AGN, leaving 281 sources.In Paper 1 we listed the number of CSOs in the three complete samples.The number of CSO 2s in each of these three complete samples is given in Table 2.The determination of a uniform set of measurements of the largest angular size of the 79 bona fide CSOs is described in Paper 1.
All the sources in the PW sample were mapped using the Cambridge 5 km Telescope by Peacock & Wall (1981), who also classified the large scale structures in the PW sample according to the following types: (i) FR I and FR II, and an intermediate FR type (FR?); (ii) objects unresolved on the 5 km Telescope (U); (iii) Compact Steep Spectrum (CSS) objects having α ≤ −0.5 between 2.7 GHz and 5 GHz; and (iv) double objects with the optical identification coincident with one of the two radio components.These types are listed in column 7 of Table 7 in the Appendix.
Discussions of a size cutoff in CSO 2s are not new (Augusto et al. 1998(Augusto et al. , 2006;;Augusto 2009), and early lobespeed measurements showed that the hotspots of CSO 2s are rapidly separating (Owsianik & Conway 1998;Owsianik et al. 1999;Polatidis et al. 2002).It was clear, therefore, as pointed out in R94 and R96, that CSO 2s must be short-lived, since otherwise there would be far more of their longer-lived, larger counterparts.This means that CSO 2s must exhibit a size cutoff.As shown in this paper, we have now determined that a sharp cutoff occurs at ≈ 500 pc.The evolution of the vast majority of CSO 2s from "early-life" through "mid-life" to "late-life" is discussed in detail in Paper 3, where we also discuss the fact that a small fraction (< 1%) of CSO 2s almost certainly go on to form the larger classes of jetted-AGN, including MSOs, FR Is and FR IIs.
Since the CSOs in the PR+CJ1+PW complete samples are all CSO 2s, the findings of this paper apply, (i) only to CSO 2s, and (ii) only to the high-luminosity end of the CSO 2 luminosity function.Much deeper complete sample surveys, in which we are engaged, are needed to expand our knowledge into the low-luminosity regime of CSO 2s.It should therefore be born in mind that our sample, comprising only 17 objects, is small, so that a degree of caution must be exercised in interpreting the results.For this reason we present all of the relevant statistics and p-values so that readers may judge for themselves the significance of the results.
There are precedents in astronomy for drawing powerful conclusions based on small numbers.For example, Hubble (1929a,b), based his discovery of the expansion of the universe on measurements of just 22 galaxies.Minkowski (1941) had just 14 supernovae for his classification of Type I and Type II supernovae, with 9 and 5 objects, respectively.Closer to the approach of this paper, there is also a powerful precedent for using well-defined statistical tests based on complete samples in the paper by Schmidt (1968), who used a complete sample of just 33 quasars to demonstrate convincingly that quasars are not evenly distributed in space, but show strong cosmological evolution.
In this paper we present three independent sets of data and lines of argument based on complete samples, each of which shows that the vast majority of CSO 2s form a distinct population of jetted-AGN.These lines of argument are based on (i) the numbers of CSO 2s in complete samples; (ii) the redshift distributions of these CSO 2s; and (iii) the size distribution of these CSO 2s.Of these, the results of first and third arguments are, in our view, compelling.The results of the second argument (ii) are significant only at the p-value=1.6× 10 −2 (2.1σ) level, and are, therefore, not compelling, but they are in the same sense as the other two arguments -i.e. they strongly suggest that the CSO 2s are drawn, predominantly, from a distinct population of jetted-AGN.
The PW sample was selected at 2.7 GHz, unlike the PR and CJ1 samples which were selected at 5 GHz.However, we have 5 GHz flux densities for all the PW sources (Pauliny-Toth 1977).Following a suggestion by John Peacock, in order to be able to combine results from these three complete samples without introducing any possible biases due to the different sample selection frequencies, we define a subset of the PW sample that is effectively complete at 5 GHz.For this purpose we compare the GB6 (Gregory et al. 1996), PR, and PW samples at 5 GHz over their common sky area (35 • ≤ δ ≤ 75 • , |b| ≥ 10 • , B1950).These surveys were all made on different instruments at different times The great strength of the PR, CJ1, and PW samples is that all of the objects are well-studied and their radio properties on both large and small scales are known.There is, therefore, no danger of unknown selection bias that could compromise the statistics.In Table 7 in the Appendix we list all of the sources in the complete PR, CJ1 and PW samples and we provide references to these structure observations.Clearly, the references given in Table 7 do not include all of the papers that refer to the objects in these samples -in many cases we provide only a single reference to a paper that contains a good map of the object.
In addition to the above three complete samples, there is one other complete sample that is of prime importance to this study: the GaLactic and Extragalactic All-Sky Murchison Widefield Array (GLEAM) survey (Callingham et al. 2017), which covers the sky area δ < 30 • (J2000), |b| > 10 • and defines a complete sample of 11,400 objects exhibiting flux densities greater than 1 Jy in the 72 MHz -700 MHz range.
In the following sections, using the complete samples, we provide three lines of argument that the vast majority of CSO 2s form a distinct population of jetted-AGN.

THE FRACTIONS OF CSO 2s IN COMPLETE SAMPLES
It is important to note that, in addition to the identification of 79 "bona fide" CSOs in Paper 1, we also identified 167 "class A" CSO candidates, which are objects showing clear double structure, but for which the maps are not of sufficient quality to confirm them as CSO 2s.We have VLBA observations of these and are in process of analyzing them.We also identified 1164 "class B" CSO candidates, most of which are far less likely to be CSO 2s, but which cannot yet definitively be ruled out.
R96 gave a detailed discussion of the CSO fractions in the PR and CJ1 complete samples.Here we update this discussion and also incorporate the PW sample.
While the PR CSO sample is complete, as can be seen in Paper 1, in CJ1 there are 5 class A CSO candidates which might possibly be bona fide CSO 2s.All of these candidates have sizes less than 500 pc, and, were we to include these five sources in our analysis, the conclusions below would be strengthened.We prefer to take the conservative route and not to include any CSO candidates in the bona fide sample until they have met the CSO criteria laid out in Paper 1.
As can be seen in Paper 1, there are also six class B CSO candidates in the PR, CJ1, and PW samples.These are much less likely to be bona fide CSO 2s, and all but one have sizes less than 500 pc.For these reasons, the conservative approach is again not to include any of these objects in the present analysis.
We see in Table 2 that the fraction of CSO 2s in complete 2.7-5 GHz samples is (6.8 ± 1.6)%.This would rise to (8.5 ± 1.8)% if all of the class A CSO 2s candidates in the CJ1 sample are shown to be bona fide CSOs.
We take as a simple hypothesis to be used throughout this paper, that, between their appearance and disappearance, the separation speed, v sep , of the opposing hot spots in CSO 2s, when averaged over a sufficient interval of time, is constant, and that they continue at the same separation speed if they expand to form larger classes of sources, such as FR IIs.Under this hypothesis, the number of objects in different size ranges scales simply in proportion to the size ranges.
It is important to note that, for the purposes of our arguments regarding the fractions of CSO 2s with respect to classes of larger sources, this hypothesis is highly conservative.We show in Paper 3 that the separation speed of the hot spots for CSO 2s is v sep = (0.36 ± 0.04)c, whereas, for example, Carilli et al. (1991) argue convincingly that for the opposing hot spots in Cygnus A, 0.01c < v sep < 0.05c.Note that this deduced separation speed for Cygnus A is typical for FR IIs (Scheuer 1995).
Based on these values, the separation speeds in CSO 2s are approximately an order of magnitude greater than those in FR IIs, which means that if CSO 2s do expand to form FR IIs, they spend far less of their time in the 0 kpc to 1 kpc size range than under the constant speed hypothesis, and so there should be even fewer of them relative to FR IIs than under the constant speed hypothesis.
Under the constant speed hypothesis we also assume that the luminosity does not change enough for the source to drop out of the flux-limited sample.We discuss possible changes in luminosity later.We consider three populations of objects that are larger than CSO 2s and that might, therefore, be the populations that CSO 2s evolve into: (i) CSS objects (Peacock & Wall 1982), including the subclass of Medium Symmetric Objects (MSOs) which have sizes in the range 1 kpc -20 kpc (Fanti et al. 1995) and R96.Note that MSOs have the same characteristics as CSO 2s apart from the size range.
Note that Fanti et al. (1995) and R96 used an upper size limit of 15h −1 kpc for MSOs, where H o = 100h km s −1 Mpc −1 .For our adopted cosmology, this translates to 21 kpc.However, since the original choice of 15h −1 kpc was chosen by Fanti et al. (1995) and R96 to be a convenient "round number", we will follow that practice and use 20 kpc as the upper size limit of MSOs in this study, which also accords with the upper limit on MSO and CSS sizes adopted by OS21.
Note-Tests #1 and #3 are independent due to their the different sky areas.We can therefore, legitimately, multiply their p-values, which we do in Test #4.While the results of this redshift test do not rise to the 3σ level, and so cannot be considered to be compelling in and of themselves, they do strongly suggest a difference between most CSO 2s and the rest of the jetted-AGN population, and are therefore supportive of the other tests we present.
The fact that the fraction of CSOs in complete samples is far too high for them all to evolve into larger classes of radio sources was first discussed by R94, and has since been much studied (see OS21).Here we discuss the results for the hona fide CSO 2s in our complete samples.We see from Table 2 that there are 19 CSO 2s and 43 CSS objects in the combined PR+CJ1+PW complete sample.Note that MSOs are a subset of the CSS class.Thus the fraction of CSO 2s in the combined CSO+CSS sample is (30 ± 8)%.Assuming an upper limit on CSS and MSO sizes of 20 kpc, on our hypothesis of constant speed of advance, the number of CSO 2s in complete samples of CSS and MSOs should be 1/20 = 5%.We therefore reject the hypothesis that a significant fraction of CSO 2s evolve into CSS+MSO sources.
The median size of the FR-I sources in the combined PR, CJ1 and PW samples shown in Fig. 1 is 180 kpc.This can be compared to the median size of the CSO 2s in these samples of 215 pc.The ratio in sizes ≈ 837, so that on the hypothesis of constant expansion speed we would expect there to be ∼15,907 FR Is, whereas there are 16 -i.e., there are ∼ 990× fewer FR Is than expected.Conversely, given the number of FR Is in these three complete samples, there are ≈ 990× more CSO 2s than expected.
The median size of the FR-II sources in the combined PR, CJ1 and PW samples shown in Fig. 1 is 305 kpc.This can be compared to the median size of the CSO 2s in these samples of 215 pc.The ratio in sizes ≈ 1420, so that on the hypothesis of constant expansion speed we would expect there to be ∼27,000 FR IIs, whereas there are 77 -i.e., there are ∼ 350× fewer FR IIs than expected.Conversely, given the number of FR IIs in these three complete samples, there are ≈ 350× more CSO 2s than expected.
We see therefore that the numbers of both FR I and FR II sources, relative to CSO 2s are far too small, by factors of over 900 for the FR Is and over 300 for the FR IIs, for CSO 2s to evolve into either FR I or FR II sources of comparable radio luminosity.At this flux density level the integrated number-flux density counts have a power-law slope of −1.3, so that the luminosity would have to drop by factors of 200 and 90, respectively to accommodate this scenario for FR I or FR II objects.Rawlings & Saunders (1991) have shown that there is a strong correlation between radio jet power and optical narrow line luminosity.Based on observations by Lawrence et al. (1996), R96 showed that the narrow line luminosities of the CSO 2s J0111+3906, J0713+4349 and J2355+4950 are about a factor 30 below that of typical FR-II galaxies, so that if CSO 2s are to evolve into FR II galaxies, then their optical line luminosities must increase by about a factor 30 while their radio luminosities decrease by about a factor 35, which seems an unlikely scenario.It is interesting to note that R96 show that the jet power for J2355+4950, when corrected for the Hubble constant and different cosmologies, is ∼ 7×10 43 erg s −1 , and for J0111+3906, and J0713+4349 the similarly corrected jet powers ∼ 10 45 erg s −1 , which may be compared to the range of jet powers in FR II sources of ∼ 10 44 erg s −1 − 10 47 erg s −1 (R96).Thus the jet powers of CSO 2s are similar to those of FR II objects, as is also the case regarding their luminosities.Given the agreement in narrow line luminosity between CSO 2s and FR I galaxies, the possible evolutionary scenario from CSO 2s to FR I galaxies may seem promising, but again, the numbers are off by over a factor 900.
We conclude on the basis of these fractions of CSO 2s in complete samples, that the vast majority (≳ 99%) of CSO 2s do not evolve into any of the above classes of larger jetted-AGN, and therefore that they belong to a distinct class of jetted-AGN.

THE REDSHIFT DISTRIBUTION OF CSO 2s IN COMPLETE SAMPLES
An independent test of whether or not CSO 2s are drawn from the same population as the other jetted-AGN in our complete samples is provided through the redshift distribution.The redshifts are listed in Table 3.The redshift distributions of the PR+CJ1 and PW complete samples, and their corresponding CSO distributions, are shown in Fig. 2.
We have carried out the Kolmogorov-Smirnov (KS) 2-sample test on the PR+CJ1 sample, the PW sample, and the PWS sample, with the results given in the four tests shown in Table 4.The cumulative distributions corresponding to Tests #1, #2 and #3, and their KS statistics, are shown in Fig. 3 (a), (b) and (c).In carrying out these tests we have removed the CSO 2s from the full samples.The KS statistic is completely determined by the data, but the corresponding p-value depends on the assumptions made in integrating over the parent distribution (Press et al. 1992).We verified that MATLAB and Numerical Recipes use the same formulae for determining the p-values.For that reason we While the observed uneven distribution could just be a result of small statistics, it would be foolish to ignore it, especially in light of the corroborating evidence from both the numbers and the redshift distributions.Nature often surprises us.
use the MATLAB p-values in deriving the significance levels in Table 4.
The first two redshift distribution tests (#s 1 and 2 in Table 4) show that the probability of the hypothesis that the CSO 2s and non-CSO 2s are drawn from the same population is 0.13 for the PR+CJ1 sample; and 0.03 for the PW sample.If we look at the effectively complete PW subsample having S 5 GHz > 1.3 Jy and at declination δ < 35 • , which is independent of the PR+CJ1 sample in view of the mutually exclusive declination limits, we see that the probability is 0.12.Since these are independent samples, we may legitimately multiply the p-values of Tests #1 and #3, which yields a probability of 1.6×10 −2 , which is significant at the 2.1σ level.While not at the 3σ level, these statistics nevertheless provide some independent evidence that CSO 2s are drawn from a different population compared to that of the other jetted-AGN in these complete samples.

Re l a t i v e Ri g h t A s c e n s i o n ( mi l l i a r c s e c o n d s )
Re l a t i v e De c l i n a t i o n ( mi l l i a r c s e c o n d s ) Figure 5. Demonstration that the complete samples studied in this paper are not restricted by the usual ∼ 100 milliarcsecond field sizes typical of most VLBI observations at 5 GHz.Shown here are 1.7 GHz VLBI maps of six large angular scale compact AGN from the CJ1 complete sample survey (Polatidis et al. 1995), all of which have sizes ≫ 100 milliarcseconds.Note that the structure of 1458+718 (J1459+7140, 3C 309.1) extends over 1 arc second -i.e., the map is ten times larger than the typical field of view of VLBI maps at 5 GHz or higher frequencies. .This result, which is seen clearly in the redshift distributions shown in Fig. 2, is interesting.If correct, it suggests that CSO 2s only started forming in significant numbers towards the end of the epoch of maximum galaxy and star formation: The lookback time to the peak in the cosmic star formation rate is ∼ 8 billion years (Förster Schreiber & Wuyts 2020), which is close to the lookback time to redshift z ≈ 0.9, when CSO 2s started to appear in significant numbers, as can be seen in Fig. 2 Thus, a possible explanation of the origin of CSO 2s is that quiescent SMBHs form CSO 2s by single star capture, and so become significant around z ∼ 1, when the numbers of both stars and SMBHs in the universe reach a maximum.We give a detailed discussion of this hypothesis in Paper 3.
However results that are significant only at the ∼ 2σ level often disappear with the advent of more data, and this particular apparent difference between CSO 2s and other jetted-AGN may disappear as more bona fide CSO 2s are accrued through new and deeper complete samples.An important point that should be mentioned here is that there are 14 bona fide CSOs in the VLBA Imaging and Polarization Survey (VIPS) of flat spectrum (α ≥ −0.5)) sources (Helmboldt et al. 2007) that are not in the complete PR+CJ1+PW samples, and of these only one has redshift greater than 1.Thus, extending the luminosity function almost an order of magnitude deeper appears not to change our findings in this section.This gives us confidence that this preliminary 2.1σ result will be greatly strengthened, when we are able to add the steep spectrum counterparts to the VIPS survey to make this a complete sample, as we are now engaged in doing.
As a distinct population, and recalling that these are all "short-lived" but not all "young" sources, it will be of great interest to investigate whether CSO 2s show the same strong cosmological evolution as do both highluminosity extended steep spectrum sources and compact flat spectrum sources (Peacock & Gull 1981), but this is beyond the scope of the present paper.

THE SIZE DISTRIBUTION OF CSOs
Our third independent test of the hypothesis that CSO 2s form a distinct class of jetted-AGN is based on the size distribution of CSO 2s.This test is more complex and more subject to selection effects than the tests of the previous two sections.Selection effects are particularly strong when it comes to consideration of the observed distribution of CSO sizes, so we discuss first the effectiveness of our approach in dealing with these selection effects, in order to give the reader some confidence in the statistical robustness of our results.

The Efficacy of Complete Samples in Dealing with the CSO Size Distribution Selection Effects
The distribution of the physical sizes of the 54 bona fide CSO 2s, out of our sample of 79, for which we have spectroscopic redshifts is shown in Fig. 4 (a).It shows a strong cutoff well below 1 kpc.However, one has to bear in mind that this sample of 54 bona fide CSO 2s is a heterogeneous sample gleaned from the literature, and is subject to selection effects.We therefore have to consider carefully whether these selection effects can be eliminated in complete sub-samples of our 54 bona fide CSO 2s.

The ∼ 100 Milliarcsecond Selection Effect
The first selection effect we consider comes about because the largest angular size that is measured in most cm-wavelength VLBI maps ∼ 100 milliarcseconds.In Fig. 4 (a) we show the upper size cutoff this would impose as a function of redshift.Only CSOs to the left of this curve have angular sizes less than 100 milliarcseconds at the corresponding redshift.Clearly this could well impose a strong selection effect on the observed size distribution of CSOs.
On the face of it, it might appear that this selection effect alone is so strong that the true size distribution of CSOs is impossible to determine from these data.Fortunately this is not the case because one can observe complete samples in which one knows the sizes of all the objects in the sample, and if some objects are too large for VLBI mapping at cm wavelengths they can be observed at longer wavelengths, where the ∼ 100 mas limit does not apply.We have availed ourselves fully of this strategy: In addition to the observations of compact objects in the PR+CJ1 complete samples at 5 GHz (Pearson & Readhead 1988;Xu et al. 1995), all of these objects were observed at 1.66 GHz (Polatidis et al. 1995;Xu et al. 1995).In Fig. 5 we show examples of six AGN from the CJ1 complete sample with sizes far exceeding the 100 mas angular size limitation of regular VLBI at 5 GHz and above.As can be seen here, even objects as large as 1 arcsecond were mapped in this survey.This is one of two reasons we can be confident that we have not missed any large CSOs in these complete samples.The other reason is that the large-, by which we mean (≳ 1 arcsec), -and small-scale radio structures of all of these objects are known.In Table 7 in the Appendix we list references which present the relevant maps of all 281 objects in the PR+CJ1+PW samples.

Spectral Shape Selection Effect
Many CSOs are peaked spectrum (PS) sources3 .Thus in a sample of CSOs selected at a single frequency, we will clearly include all of the sources that peak at that frequency down to the flux density limit.However, for sources that peak at frequencies significantly higher or lower than the selection frequency, the sample will exclude an increasing number of the CSOs as the separation between the peak frequency and the selection frequency increases.In this study, we therefore consider not only the PR+CJ1 and PW samples, selected at 5 GHz and 2.7 GHz, but we also consider the GLEAM sample, observed using 20 simultaneous flux density measurements spanning frequencies between 72 MHz and 231 MHz, the 3CRR sample selected at 178 MHz, and the Jodrell Bank 966 MHz sample.
The situation is illustrated in Fig. 6.The blue points show the observed radio spectrum of OQ 208 (J1407+2827) (Stanghellini et al. 1997), which has one of the narrowest, most sharply peaked, spectra amongst the bona fide CSO 2s.The gray points illustrate an object with the same spectral shape as OQ 208, but with the maximum shifted from 5 GHz down to 1 GHz, and the peak flux density shifted down to 1.3 Jy.This is the point where the object would drop below the GLEAM 1 Jy limit (Callingham et al. 2017), and the CJ1 700 mJy limit.Because of the drop-off in flux density, relative to the peak, at both higher and lower frequencies, such  an object would not be included in the PW, PR, CJ1 or GLEAM samples.Objects of this type with peak flux densities greater than 1.3 Jy would, however, be included in the GLEAM and CJ1 samples, whose limiting flux densities are indicated by the horizontal brown line in Fig. 6, and the red horizontal bar, respectively.We therefore consider next what is known about the population of objects that peak at frequencies ≲ 1 GHz.
The GLEAM survey covers 72 MHz -700 MHz, and is concentrated in the southern hemisphere, and so overlaps only part of the sky area covered by the PR+CJ1+PW sample, but it complements the higher frequency VLBI surveys that have studied complete samples because of its lower frequency.505 of the 11,400 sources in the complete 1 Jy GLEAM sample ((4.4 ± 0.2) %) are peaked-spectrum objects (Callingham et al. 2017), with peak flux densities above 1 Jy in the 72 MHz -700 MHz range, and so must also be compact (Readhead et al. 2021).Similarly, in a LO-FAR study of northern radio sources at 150 MHz, Slob et al. (2022) found 373 PS sources and concluded that ∼ 2.5% of sources in complete samples around 150 MHz are PS sources.We note the similarity in the fractions of PS sources identified in the LOFAR (150 MHz), GLEAM (72 MHz -700 MHz), and the fraction of PW+CJ1+PW (2.7 GHz and 5 GHz) CSO 2s samples, which are ∼ 2.5%, ∼ 4.4%, and ∼ 6.8%, respectively.It is possible that a significant fraction of the PS population in both the LOFAR and GLEAM surveys are CSO 2s, and thus that there is a significant population of CSO 2s extending to below 100 MHz.In this case we could well be missing CSO 2s that could fall into in the 500 pc -1 kpc size range.In Fig. 7 we show the sizes of the PR+CJ1+PW CSO 2s plotted against peak frequency.It is interesting to note that almost half of these objects have peak frequencies in the range covered by the GLEAM survey, even though they were selected at 2.7 GH and/or 5 GHZ.It is also interesting to note that, apart from the small fraction (< 25%) that has peak frequencies above 3 GHz, there is no clear dependence of size on frequency.

A Spectral Shape Lacuna
As we have seen in previous sections, we are only considering the 17 bona fide CSO 2s in the PR+CJ2+PW complete sample with known spectroscopic redshifts, and there are only two bona fide CSOs in these complete samples without a spectroscopic redshift.The PR+CJ2+PW complete sample, excluding M82, consists of the 281 sources listed in Table 7, including M82.In 5.1.2we discussed a spectral shape selection effect that can be affecting this sample.The GLEAM survey detected 11,400 sources with flux densities greater than 1 Jy between 70 MHz and 700 MHz.Of these 505 are PS sources.In order to double the numbers of CSO 2s, and hence potentially to have a strong effect on any statistical tests of the size distribution of CSO 2s, there would need to be ≈17 bona fide CSO 2s in the GLEAM sample.Thus only a small fraction ∼ 3% (17/505) of the PS sources in the GLEAM would need to be CSO 2s in order potentially to have a significant impact on the statistics.So this is a lacuna that has to be addressed in any size tests.
In the next three subsections we advance two independent arguments to address this lacuna and we suggest a test that could fill the lacuna, but which requires more observations and is therefore beyond the scope of the present paper.

The Range of Peak Frequencies in Our Sample
In Paper 1, Fig. 6, we have plotted the range of peak frequencies observed, and it can be seen that the peak frequencies range from below 80 MHz to ∼ 10 GHz.The same is true of the objects in our combined PR+CJ1+PW sample -the lowest peak is at 70 MHz and the highest peak is at 6.4 GHz, and the distribution of the peaks is roughly uniform between these extremes.
Thus the selection procedure of the complete PR+CJ1+PW sample and our bona fide CSO identification method do not appear to have created a bias against CSO 2s peaking anywhere within this range.However, while the (rarer) flat-spectrum CSO 2s will not suffer from the spectral selection biases described earlier, some peaked-spectrum CSO 2s could be excluded from the sample for certain redshift ranges.This could, therefore, influence the size distribution of the observed CSO 2s in the PR+CJ1+PW samples, particularly if CSO intrinsic size is related to peak frequency and/or luminosity.
In the next two subsections we give an argument that shows that spectral shape selection effects are unlikely to have biased the size distribution of the CSO 2s in the PR+CJ1+PW sample.

The 3CRR and PW CSS Double Sample
In addition to our complete samples of CSO 2s, described in the previous sections, there is one other relevant sample of CSO 2s and MSOs that has been studied extensively by the Bologna Group (BG), the key results of which are given in a series of papers (Fanti et al. 1985(Fanti et al. , 1990;;Spencer et al. 1991;Fanti et al. 1995;Dallacasa et al. 1995Dallacasa et al. , 2013Dallacasa et al. , 2021)).The BG identified 32 doublelobed CSS objects, given in Table 1 of Fanti et al. (1995), in their sample drawn from the 3CRR (Laing et al. 1983) and the PW samples.They subsequently added one double-lobed source (1819+396 = 4C +39.56) that they had previously missed (Dallacasa et al. 2021), bringing the total of double-lobed CSS sources in the BG sample to 33.Given that these are CSS objects, they excluded flat spectrum objects with α > −0.5.
This spectral filter against flat spectrum sources (α > −0.5), as applied to the 3CRR sample, which has a limiting flux density of 10 Jy at 178 MHz, excludes sources brighter than 2.57 Jy at 2.7 GHz. Since these are greater than the flux density limit of the PW complete sample (1.5 Jy at 2.7 GHz), any such object can be included in the BG study by adding the flat spectrum PW CSO 2s to the steep spectrum CSO 2s detected by the BG, in order to get the total of both flat and steep spectrum compact doubles, including CSO 2s, in the 3CRR and PW samples (note that there are no flat spectrum doubles of size greater than 1 kpc in PW).There are four BG CSOs in our sample of 79 bona fide CSOs listed in Paper 1.All four of these BG CSOs are already in our complete sample of CSOs in the PR+CJ1+PW complete samples.
We therefore compensate for the spectral index limit in BG sample by adding the flat spectrum PW bona fide CSO 2s that were excluded from the BG sample by the spectral index cutoff at α = −0.5 to the BG sample of CSO 2s, thereby making this into a complete sample of 3CRR+PW compact double sources, and bringing the total including the PW flat spectrum CSO 2s to 40.In order to apply the same largest angular size filter as that used in Paper 1, we have re-measured the largest angular sizes of the 33 BG CSS double sources at the lowest frequency at which high-quality maps are available in the BG group's publications listed above.The results are shown in Fig. 8, together with the 7 flat spectrum PW bona fide CSO 2s that we have added.
We find that 27 of the 33 objects in the BG sample fit the CSO+MSO criteria, with four of the objects being bona fide CSO 2s and the remaining 23 objects being MSOs.The 6 remaining objects all have largest projected physical sizes greater than 20 kpc, based on our measured largest angular sizes.When we add the flat spectrum objects from the PW sample, the number of CSO 2s increases from 4 to 11, as shown in Fig. 8.
The four CSO 2s in the BG CSS sample all have sizes between 300 pc and 500 pc.These PS CSO 2s have spectral turnovers that are almost certainly due to synchrotron self absorption as is shown in the paper by Scott & Readhead (1977), who showed that the equipartition angular size ψ eq ∝ S .Thus fainter objects that show spectral peaks at higher frequencies will be smaller than the four CSS CSO objects in the BG sample.It therefore is unlikely that there will be a significant number of CSS CSO 2s with sizes in the 500 pc to 1 kpc range.We return to this point in §5.3.   5.

The Jodrell Bank 966 MHz Sample
. Since the 3CRR sample is complete down to 10 Jy at 178 MHz (Laing et al. 1983), for comparison with the other samples in this study it would be helpful to have a low frequency sample complete down to ∼ 1 Jy.Fortunately, such a sample exists for which the radio structures of over 98% of the objects are known.
Referring back to Fig. 6 and the objects in the lacuna illustrated by the gray spectrum.We can define a complete sample drawn from the Jodrell Bank 966 MHz survey (Cohen et al. 1977), which produced a radio catalogue and measured arcsec-level positions for the majority of its sources.We have selected a sub-sample, consisting of 169 of the strongest sources (S 0.966 > 1.5 Jy) from Cohen et al. (1977).This sub-sample is unbiased, and while the full survey is not strictly complete due to confusion issues, these apply only at low flux density levels, and thus the sub-sample that we have selected is not affected by confusion.We will refer to this unbiased sub-sample of 169 objects as the "JBS" sample.
We classified 74 of the JBS sample in the filtering process we carried out in selecting our bona fide CSO 2s described in Paper 1. We identified six of them as bona fide CSO 2s.
We have extracted VLASS cutout images of all 169 JBS objects using the CIRADA cutout server4 , and we found only 17 of them to be unresolved, with largest angular size < 3 arc sec, and hence possible CSO 2s.Of these 17 compact objects, two are MOJAVE "core-jet" objects, and one is a 2 arc second double.Thus there are 14 possible CSO 2s in the 966 MHz JBS sample in addition to the 6 bona fide CSO 2s we have already identified.We are engaged in obtaining VLBI observations of these 14 objects.As we have seen, all of the bona fide CSOs in the complete PR, CJ1 and PW samples are CSO 2s -i.e., as discussed in Paper 1, they are edge-brightened, highluminosity objects.This is a selection effect resulting from the flux density limits in the complete samples.The size distribution of the CSO 2s in the PR+CJ1+PW complete sample is shown in Fig. 4(b), binned into 100 pc and 500 pc intervals.

Statistical Analysis of CSO Sizes in the Complete Samples
Using the CSO 2 size distributions in the PR+CJ1 and PW complete samples, we have carried out two sets of statistical tests of the hypothesis that CSO 2s are uniformly distributed in size between 0 pc and 1000 pc, as would be expected on the hypothesis that the speed of advance is constant: (i) a set of KS 1-sample tests, which yield the cumulative distributions shown in Fig. 9 (a),  (b) and (c) and the p-values shown in Tests #5 -#8 in Table 5; and (ii) binomial tests in which we divided the CSO 2s into two size bins, from 0 pc to 500 pc, and from 500 pc to 1000 pc, which yield the results shown in Table 6.
We consider first the KS tests shown in Table 5.We see there that the uniform hypothesis is rejected by the PR+CJ1 CSO sample at the 9.3×10 −3 probability level, and by the PW CSO sample at the 8.7 × 10 −4 probability level.The independent, effectively complete, PW CSO sample below declination 35 • (Test #7) rejects the uniform hypothesis at the 1.7 × 10 −2 level.
Tests #5 and #7 are independent -note the different sky areas -and at the same observing frequency.We can therefore, legitimately, multiply their p-values.This gives Test #8, which rejects the uniform size distribution hypothesis at the p-value 1.6×10 −4 , or 3.6σ significance level.
The results of the binomial tests, shown in Table 6, show that by similarly combining the PR+CJ1 CSO sample (Test #9) with the independent PWS sample (Test #10), as shown in Test #11, the uniform hypothesis is rejected at the 1.7 × 10 −4 (3.6σ) level.
In our view, these tests on complete samples constitute compelling evidence that the size distribution of CSO 2s cuts off sharply at ≈ 500 pc, which is significantly below the 1 kpc size limit imposed by the defining criteria of CSOs.As described in §5.2.3, the existence of this sharp cutoff can be tested, for example, with MER-LIN and VLBI observations of the 14 potential CSO 2s in the complete JBS sample.
Clearly some CSO 2s must grow to larger sizes in order to produce MSOs.An example of a CSO and MSO with remarkably similar morphologies is shown in Fig. 10, which illustrates this point, especially since 0404+768 (J0410+7656) was originally classified as a CSO, but  (1995), and (d) from Taylor et al. (1996): 1358+624 (J1400+6210).Some CSO 2s must evolve into MSOs.and 1358+624 may be just such a case.The red cross marks the location of the core in each map.
. fails the size cutoff (by ∼20%).It is clear that the majority of CSO 2s do not grow much above 500 pc in projected size.

RESULTS AND TESTS FROM THIS PAPER
The results and tests that have come out of this study are listed below.

Results
The number fraction, redshift, and size statistics discussed in the previous three sections and presented in Tables 4 to 6 demonstrate that (i) most CSO 2s are drawn from a population of jetted-AGN that is distinct from other jetted-AGN, and (ii) the size distribution of CSO 2s cuts off sharply at ≈ 500 pc, a finding that is verifiable (see §5.2.3).
These are significant findings.They show that there has to be something fundamentally different between CSO 2s and the larger sources.While both must be driven by the same type of central engine, since both are producing high-luminosity relativistic jets, there must be some fundamental difference between them to produce two such different outcomes -one with a size cutoff around 500 pc and the other with a size cutoff ∼ 200× larger.
One might think, for example, that the cutoff could be explained simply by random episodic fuelling.But how would random episodic fuelling produce a sharp cutoff?Random episodic fuelling would produce a uniform distribution.There has to be another explanation for the cutoff, such as, for example, an upper limit on the size of the fuel packages, a change in the jet environment that leads the jets to fade beyond a certain distance from the central engine, or a mechanism associated with the accretion disk that limits the energy of the jet.

Tests
The major tests resulting from this paper are the following: 1. We will follow-up the JBS sample with MERLIN and VLBA observations in order to test the sharp cutoff in size we have seen in the PR+CJ1+PW complete sample.
2. We have undertaken a program to increase the number of CSO 2s in complete samples by a factor of at least 3, to ∼ 50 by carrying out a VLBI survey of ∼ 332 steep spectrum sources that complement the VIPS flat spectrum survey, thus converting VIPS into a complete sample.This will further test the cutoff in CSO 2 size distribution.
3. Structural studies of the PS objects identified in the LOFAR and GLEAM surveys would be well worth doing.

DISCUSSION
Although the selection effects inherent in our literature search for CSOs are significant, we have shown that, with careful use of complete samples, and using the 17 CSO 2s in the complete PR, CJ1, and PW samples for which we have spectroscopic redshifts, it is possible to carry out a series of rigorous statistical tests that provide what we regard as compelling evidence that the vast majority of CSO 2s constitute a population of jetted-AGN that is distinct from, and therefore requires a separate origin to, the larger classes of jetted-AGN, such as CSS sources, MSOs, and FR I and FR II objects.
The physical size cutoff is clearly telling us something important about this class of jetted-AGN.The scenario that produces almost all CSO 2s must be different in some important way from that which produces the larger symmetric radio sources.We return to this discussion of the origins of CSO 2s in Paper 3.
It should be clear, therefore, that, (i) because the observed emission regions in these objects are not significantly relativistically boosted towards the observer, thereby making it possible to determine their detailed physical properties, and (ii) most CSO 2s belong to a distinct class of jetted-AGN, these CSO 2s provide a unique time domain plus structural window on the central engines of jetted-AGN and the supermassive black holes that drive them.from the Foundation of Research and Technology -Hellas Synergy Grants Program through project POLAR, jointly implemented by the Institute of Astrophysics and the Institute of Computer Science.A.S. was supported by the NASA Contract NAS8-03060 to the Chandra Xray Center.

Figure 1 .
Figure 1.The distributions of the largest projected sizes of FR I objects (top panel) and FR II objects (bottom panel) in the PR+CJ1+PW complete samples.The FR I sizes have been determined from our own angular size measurements.The FR II sizes are based on the largest angular size measurements ofNilsson et al. (1993) for all but six sources not included in their sample, for which we measured the angular sizes ourselves.There is one FR II object (3C 236) of size 4.3 Mpc that is not included in the FR II plot.

Figure 2 .
Figure 2. The redshift distributions for the PR+CJ1 complete sample (top panel) and the PW complete sample (bottom panel).The light shaded distributions show the complete samples.The dark shaded regions show the CSO 2s.Note that these distributions are not stacked vertically, so the values on the ordinate represent the total numbers of sources and the numbers of CSO 2s in each sample.The cumulative distributions and KS statistics are shown in Figs. 3 (a) and (b).

Figure 3 .
Figure 3. KS Tests on the redshift distributions of the bona fide CSO 2s in the PR+CJ1, PW, and PWS samples.(a), (b) and (c): comparison of the CSO cumulative redshift distributions vs. the non-CSO 2s in the complete PR+CJ1, PW, and PWS samples, respectively.The green bars indicate the maximum differences in the cumulative distributions, corresponding to the values of the KS statistic given by the numbers in green.The corresponding p-values are listed in Table4.

Figure 4 .
Figure 4.The distributions in size of the bona fide CSOs over the whole CSO size range, from 0 pc to 1 kpc: (a) The heavy black boxes show the histogram of the sample of 54 bona fide CSOs for which there are spectroscopic redshifts, with the numbers given on the left axis.The dashed curve, marking the border of the shaded region, shows the physical size corresponding to 100 milliarcseconds at the redshift indicated on the right-hand axis.For typical VLBI observations at 5 GHz and above, CSOs in the grayed region to the right of this curve would be hard to observe, so there is a strong selection effect that might account for the drop in numbers of bona fide CSOs with physical size.(b) The 17 bona fide CSO 2s with spectroscopic redshifts in the complete flux densitylimited PR+CJ1+PW sample.Dotted curves show the data binned into 100 pc bins, while solid curves show the data binned into 500 pc bins.The Kolmogorov-Smirnov and binomial tests both show that this distribution differs from a uniform distribution at the p-value ∼ 1.7 × 10 −4 (3.6σ) level.While the observed uneven distribution could just be a result of small statistics, it would be foolish to ignore it, especially in light of the corroborating evidence from both the numbers and the redshift distributions.Nature often surprises us.
(a) and (b).The peak Star Formation Rate (SFR) occurs from z ≈ 1 to z ≈ 2, with the peak Supermassive Black Hole (SMBH) formation rate (Tacconi et al. 2020) peaking slightly after the peak SFR.

Figure 6 .
Figure 6.CSO 2s that might be missed in the PR, CJ1, and PW complete samples due to spectral effects: The blue, green and red arrows indicate the selection frequency and limiting flux densities of the PW, PR, and CJ1 samples, respectively.The horizontal brown line indicates the flux density limit of the GLEAM sample.The blue and gray points show the observed spectrum of OQ 208 (Stanghellini et al. 1997), and a shifted spectrum of OQ 208, respectively (see text).

Figure 7 .
Figure 7.The relationship between size and peak frequency for the CSO 2s in the PR+CJ1+PW complete sample.

Figure 8 .
Figure 8.The distributions of the bona fide CSO 2s and MSOs as a function of projected physical size in the BG sample.The red distribution shows the BG CSO 2s+MSOs+CSS>20 kpc objects.The green distribution shows the BG+PW CSO 2s.

Figure 9 .
Figure 9. KS Tests on the size distributions of the bona fide CSO 2s in the PR+CJ1 and PW samples.(a), (b) and (c): comparison of the CSO 2 cumulative size distributions vs. the uniform model for the PR+CJ1, PW, and PWS samples, respectively.The green bars indicate the maximum differences in the cumulative distributions, corresponding to the values of the KS statistic given by the numbers in green.The corresponding p-values are listed in Table5.

Table 1 .
The CSO Samples Note-ThisTable shows the numbers of bona fide CSOs of classes 1 and 2 identified in Paper 3, with and without spectroscopic redshifts.Numbers in parentheses indicate bona fide CSOs in the PR+CJ1+PW complete samples.

Table 2 .
The Numbers of CSS, FR I, FR II, and CSO objects in the Complete Samples.Note-All of the CSOs in the PR+CJ1+PW sample are CSO 2s.In addition, all of the PW CSOs at δ ≥ 35 • are in the PD+CJ1 sample.† the numbers exclude 3C 231 (M82), a starburst galaxy, not an AGN.‡ the numbers include the bona fide CSO J1335+5844, for which there is no published spectroscopic redshift.*thenumbers include the bona fide CSO J1416+3444, for which there is no published spectroscopic redshift.These numbers are taken from the list of the 282 sources in the three complete samples given in Table7in the Appendix.PWS is the subsample of PW at 10 • < δ < 35 • (B1950) and with S5 GHz > 1.3 Jy.As should be clear in view of the size of the samples, and assuming there is no dependence of the CSO fraction on flux density, the most reliable statistic is the final one combining the three full samples PR, CJ1, and PW.

Table 4 . . Table 4 .
Two-sample KS Tests of CSO Redshifts as a Distinct Population

Table 5 .
One-sample KS Tests Against a Uniform Distribution of CSO 2 Sizes Tests of the observed CSO 2 size distribution compared to a uniform size distribution.The PWS sample is the effectively complete subsample the PW sample having 10 • < δ < 35 • , |b| > 10 • and S5 GHz ≥ 1.3 Jy (see text).

Table 6 .
Binomial Tests of Significance Levels of CSO Size Distributions in Complete Samples Note-The PR+CJ1 and PWS (δ < 35 • , S5 > 1.3 Jy) samples are independent, so we have multiplied their p-values in Test 11 (see text).