The hierarchical clustering method: abundance and properties of local satellite populations

ABSTRACT

The faint satellites of the local Universe provide an important benchmark for our understanding of structure formation and galaxy formation, but satellite populations are hard to identify beyond the Local Group. We recently developed an iterative method to quantify satellite abundance using galaxy clustering and tested it on a local sample in the COSMOS field, where accurate photometric redshifts are available for a large number of faint objects. In this paper, we consider the properties of these satellite populations in more detail, studying the satellite stellar mass function (SSMF), the satellite-central connection, and quenching as a function of satellite and central mass and colour. Despite the limited sample size, our results show good consistency with those from much larger surveys and constrain the SSMF down to some of the lowest primary masses considered to date. We reproduce several known trends in satellite abundance and quenching, and find evidence for one new one, a dependence of the quiescent fraction on the primary-to-secondary halo mass ratio. We discuss the prospects for the clustering method in current and forthcoming surveys.

1 INTRODUCTION

On large scales, the visible structure of the universe is clearly dominated by cold dark matter (CDM), or something that behaves very much like it. Given this framework, galaxies are predicted to form and grow in the centres of dense, virialized regions known as CDM haloes. Haloes themselves grow both by the gradual accretion of diffuse matter and by stochastic mergers with other haloes. During these mergers, smaller haloes falling into larger ones will often survive as distinct, long-lived substructures within the final system. The central galaxies of these smaller haloes will correspondingly survive as distinct ‘satellite’ galaxies, orbiting the central galaxy of the larger halo. Dark matter substructure is intrinsically hard to detect directly, so satellite galaxies provide an important tracer of small-scale structure and halo assembly.

Galaxy formation is clearly strongly modulated on these small scales; a number of feedback mechanisms related to internal star formation (e.g. Dekel & Silk 1986; Mashchenko, Wadsley & Couchman 2008; Governato et al. 2010; Wetzel et al. 2016), the photoionizing background (e.g. Bullock, Kravtsov & Weinberg 2000; Gnedin & Kravtsov 2006; Lunnan et al. 2012; Katz et al. 2020), and/or environmental effects (e.g. Taylor & Babul 2001; Mayer et al. 2006; Łokas, Kazantzidis & Mayer 2012; Richings et al. 2020) probably combine to greatly reduce the abundance of dwarf satellites, but it is possible dark matter structure itself is modified or suppressed on these scales (e.g. Spergel & Steinhardt 2000; Macciò & Fontanot 2010; Anderhalden et al. 2013; Kennedy et al. 2014; Lovell et al. 2014; Elbert et al. 2015; Fry et al. 2015; Nadler et al. 2019). Satellite populations provide an important test of this rich array of physical processes.

The Local Group (LG), a composite system dominated by the Milky Way (MW) and the Andromeda Galaxy (M31), has the best-studied satellite populations. We can detect LG satellites down to much fainter magnitude and surface brightness limits than in any external group, and can resolve systems into individual stars, allowing detailed studies of their evolutionary history (McConnachie 2012). It is not clear, however, that the LG is completely representative of the group environment or of satellite populations. Observational studies have shown that it is relatively rare for galaxies like the MW to have two bright star-forming satellites like the Magellanic Clouds, for instance (Guo et al. 2011; Liu et al. 2011; Robotham et al. 2012; Strigari & Wechsler 2012; Speller & Taylor 2014), and numerical simulations have reached similar conclusions (Boylan-Kolchin et al. 2010; Busha et al. 2011; Kang, Wang & Luo 2016; Zhang, Luo & Kang 2019; Evans et al. 2020). Complete spectroscopic surveys (e.g. Geha et al. 2017; Mao et al. 2021) are the ultimate tool for extending our understanding of satellite populations to other groups, but they can be very expensive in observing time, particularly for low-mass systems where genuine satellites are rare.

In previous work (Speller & Taylor 2014; Xi et al. 2018), we have explored the use of galaxy clustering to quantify satellite populations around isolated primaries. In Xi & Taylor (2021; ‘paper I’ hereafter), we developed and tested a new, hierarchical method for quantifying satellite abundance from galaxy clustering measurements (the ‘clustering method’ hereafter). The method uses the most obvious central galaxies, those that clearly dominate a volume in projected area and redshift around them, to establish a basic clustering template for the radial dependence and amplitude of the satellite population. This template is then applied iteratively to the entire survey, assigning probabilities that any given galaxy is a satellite of a given nearby primary galaxy. The clustering method is somewhat similar to the group-finding methods of Yang et al. (2007) and Kourkchi & Tully (2017), which also use adaptive search radii for group members, and iteratively update satellite membership and group properties. Our method does not assign definite membership for each satellite, but only estimates the probability of each galaxy pair being associated as primary and secondary. The method is particularly useful in crowded fields, or where distance information is limited.

We tested the method using data from the low-redshift part of the COSMOS field. COSMOS is a deep (AB ∼ 25–26), multiwavelength (0.25–24 µm) survey covering a 2 deg² equatorial field (Scoville et al. 2007). The data available for the COSMOS field include high-quality photometric redshifts (photo-zs) generated from the 30+ deep bands (Scoville et al. 2007; Ilbert et al. 2013; Laigle et al. 2016; Weaver et al. 2022), providing distance information for a large sample of faint galaxies below the spectroscopic limit. The combination of depth and photo-z accuracy makes this data set an ideal test case for the clustering method. In COSMOS, we made significant detections of the satellite population over a wide range of primary mass (∼10¹⁰–10^13.5M_⊙ in halo mass). We measured the overall satellite abundance and the satellite luminosity function, as well as their dependence on primary halo mass. We also tested for a large number of possible systematic uncertainties and showed that these are generally within the derived statistical uncertainties.

In this paper, we explore the properties of the detected satellite populations further. The outline of this paper is as follows. First, in Section 2, we review the clustering method and define the base catalogue, the first-run primary sample, and the primary and satellite probabilities used in subsequent calculations. In Section 3, we measure the satellite stellar mass function (SSMF), both in absolute terms and relative to the stellar mass of the primary. In Section 4, we study satellite abundance as a function of primary colour or specific star formation rate (sSFR, the star formation rate per unit stellar mass) and discuss the evidence for ‘conformity’ in our spectroscopic subsample. In Section 5, we measure the quiescent fraction as a function of secondary stellar mass, primary stellar mass, primary colour, and primary-to-secondary halo mass ratio. Finally, we summarize our results and discuss future prospects for the clustering method in Section 6.

2 REVIEW OF THE METHOD

2.1 The hierarchical clustering method

To characterize satellite populations, we need to measure the abundance of satellite (or ‘secondary’) galaxies around central (or ‘primary’) galaxies. This is challenging, as satellites are faint and relatively rare with respect to foreground and background galaxies seen in projection. Our method uses projected clustering, relying on the fact that satellites are located close to centrals, and thus cause a slight density excess in the projected galaxy distribution. This overdensity can be used to estimate the satellite abundance statistically, giving each galaxy a probability of being associated with a given central, based on the measured clustering strength.

In paper I, we tested our method using data from the COSMOS field. Starting with the COSMOS2015 photometric redshift catalogue (Laigle et al. 2016), we applied the following cuts to select a low-redshift sample of galaxies with reasonably accurate redshift information:

• i⁺ < 25.5;

• 0 < z_pdf < 6.9;

• z − 2σ_z < 0.3;

• σ_z < 0.5.

These cuts produce a base catalogue of 41 559 galaxies that we use to do the initial ‘first-run’ primary–secondary separation. After this step is complete, we apply a further cut σ_z < 0.25 to exclude galaxies with large redshift errors, reducing the final catalogue to 37 578 objects. In most of our analysis, we also apply a stellar mass cut at M_* > 10^7.3 M_⊙, as our sample is reasonably complete above this mass (see section 2.4 of paper I).

To define an initial sample of primaries, we make a conservative selection of systems most likely to dominate their surrounding region. First, a halo mass is assigned to each galaxy in the base catalogue as if it were the central galaxy of its own host halo, using a mean halo-to-stellar mass ratio based on Behroozi, Wechsler & Conroy (2013; see paper I, section 3.1), and the virial radius R_vir and circular velocity at the virial radius v_vir corresponding to this halo mass are calculated. Each galaxy, in order of the assigned halo mass from high to low, is then checked to see if it is the most massive galaxy within its local region of interest (ROI), and thus a potential primary.

As discussed in paper I, the ROI is a cylindrical region around each primary, extending out to a maximum radius AR_vir in projected separation R_P (calculated at the distance of the primary) and a length ±Bv_vir along the velocity-difference axis Δv. In addition, since velocity uncertainties often exceed v_vir, we put an additional requirement of Δv < Cσ_Δv on the sample of galaxies within the ROI, where σ_Δv is the uncertainty in velocity difference Δv. Based on the clustering signal measured in stacked samples, we choose the values (A, B, C)  = (3.0, 2.0, 1.5) to define the limits of the ROI, as discussed in section 3.2 of paper I.

Primary candidates close to the boundaries of the survey volume in position or in redshift, or near heavily masked regions, could have more massive companions that were not included in the catalogue. Thus, we apply two additional cuts to the list of potential primaries to ensure reasonably complete coverage of the ROI around each one:

• z < 0.25;

• area completeness >0.65.

This selection produces an initial sample of 815 galaxies, which we will refer to as the ‘first-run’ primary sample. All other galaxies in the catalogue at this point are considered potential secondaries.

We measure the clustering signal around the first-run sample of primaries and model it as the sum of two components: a satellite component and a background component of constant surface density. We assume that the background component depends only on redshift, while the satellite component depends only on halo mass (given the fairly narrow redshift range of our sample). Fitting this two-component model to the data, we can estimate what fraction of all galaxies at a given distance from the primary are genuine satellites. Thus, in a second step, for every secondary in the ROI of a primary i, we calculate this probability |$P_{\rm sat}^i$| of being a satellite. (We will also refer to the converse ‘field’ probability, |$P_{\rm field}^i = 1- P_{\rm sat}^i$|⁠, that a galaxy is not a satellite of primary i.) Given that the probability |$P_{\rm sat}^i$| depends both on the secondary and on the primary, we will also refer to primary–secondary pairs, and associate with each a probability |$P = P_{\rm sat}^i$|⁠. We note that since ROIs can overlap, a given secondary may be a member of more than one pair, having a non-zero probability of being a satellite of more than one nearby primary. Thus, the total number of pairs exceeds the number of objects in the base catalogue.

Fig. 1 shows the number of pairs versus satellite probability (lower red histogram), as well as the cumulative number over a given probability, with (upper black histogram) and without (middle blue histogram) the spectroscopically confirmed pairs discussed in Section 2.3, which we assume to be 100  per cent probable (point on the right-hand axis). We note that the pair distribution is very close to a power law in probability p, n(p) ∝ p⁻².

For a galaxy to be a primary, it must not be a satellite of any other system. Thus, if a secondary is a member of more than one pair, then treating the probabilities |$P_{\rm field}^i$| as independent, the net primary or field probability P_field is the product of the individual probabilities:

Similarly, the net probability of a galaxy being a satellite of some system, P_sat, is:

The initial primary selection is deliberately very conservative, including only those galaxies most likely to dominate their ROI. Given the non-zero background at all radii, however, no secondary is identified as a satellite with 100  per cent probability, and thus any galaxy in the catalogue could be a potential primary. To account for this, we iterate, measuring the clustering signal stacked around all objects in the base catalogue, but weighting the contribution of each galaxy by its primary (i.e. field) probability. This iteration is particularly important at low masses, as these galaxies are mainly excluded from the first-run primary sample (see fig. 3 of paper I; the first-run sample includes almost no objects below log (M_h/M_⊙) = 10.5, whereas the base catalogue extends down to log (M_h/M_⊙) ∼ 10).

2.2 Calculating distributions of satellite properties

The satellite probabilities described above allow us to study statistically the distribution of ancillary properties such as colour, luminosity, stellar mass, or star formation rate (SFR). The simplest approach, which we called ‘Method A’ in our previous paper, is to use the satellite probabilities as weights when calculating the distribution of a given ancillary property. Thus, for instance, we could estimate the satellite colour distribution by calculating the colour distribution for the entire base catalogue, weighted by the satellite probability. This assumes, however, that the ancillary property in question (colour, in this example) is uncorrelated with the clustering, which may not be true in many cases. Results calculated using Method A can be biased if the unclustered background population has a distribution of ancillary properties different from that of the satellite population (cf. appendix C of paper I); in this case, selecting on the ancillary property first will change the clustering strength.

Ideally, we would resolve this problem by splitting the catalogue into narrow bins in the ancillary property, and recalculating the clustering signal and satellite probabilities for each of these bins individually. (We called this ‘Method C’ in the previous paper.) Given the limited signal-to-noise ratio (SNR) of the clustering signal, however, binning our sample to this degree would increase the Poisson noise in the background estimate to unacceptable levels. In paper I, we described a practical alternative (‘Method B’) that works as follows: The total abundance of the satellite and background components within the ROI is estimated from the satellite probabilities described above. We also use these probabilities to define ‘field’ galaxies, with net probabilities of being satellites of less than 1  per cent, throughout the entire survey. This field population is used to define a (normalized) background distribution in the ancillary property, e.g. colour, stellar mass, etc., and it is assumed the background component in the ROI follows this same distribution. Thus, we can calculate the background contribution to a given bin in the ancillary property as the fraction of field galaxies in that bin times the total number of background galaxies in the ROI [see sections 5.1 and 5.2 of paper I, as well as Speller & Taylor (2014), for further discussion].

2.3 Spectroscopically confirmed subsample

A total of 2506 spectroscopic redshifts are also available for the base catalogue. Searching through this sample, we find 278 spectroscopically confirmed primary–secondary pairs with projected separations of less than 2.0 times the virial radius of the primary and redshift separations of less than 10⁻³ (roughly 300 km s⁻¹). This subsample includes 136 unique primaries and 271 unique satellites (by the definitions above, there are a few objects that could be satellites of more than one primary). On average, each primary in the spectroscopic subsample has slightly more than two confirmed satellites, but the most massive primaries contain significantly more satellites than the rest. There are 15 primaries with log (M_*/M_⊙) > 10.9, with a total of 65 confirmed satellites (or 4.3 each on average). We will treat the 278 primary–secondary pairs as completely certain, and use this subsample to test some of our results. We note, however, that it is assembled from many different redshift sources and is not complete nor homogeneous.

The spectroscopic sample provides an interesting test of our estimated satellite probabilities. The clustering method identifies 442 (130) secondaries with P_sat larger than 0.3 (0.5), of which 26 (14) have spectroscopic redshifts and 7 (2) are spectroscopically confirmed satellites. This is completely consistent with expectations for the P_sat > 0.3 sample (46  per cent probability of getting 7/26  = 27  per cent or fewer confirmed satellites), but inconsistent for the P_sat > 0.5 sample (only a 0.64  per cent probability of getting 2/14  = 14  per cent or fewer confirmed satellites, corresponding to a 2.7σ deviation.) Given the small numbers of objects involved, however, it is unclear whether the latter discrepancy is important, or simply an artefact of uneven spectroscopic follow-up (e.g. with coverage that undersamples close pairs).

We note that the high-probability samples provide an interesting set of possible targets for further spectroscopic follow-up; it would be fairly easy, for instance, to obtain spectra for magnitude-limited samples of the (442 − 26)  = 416 objects with P_sat > 0.3, or the 2008 objects with P_sat > 0.1, with the expectation that significant fractions of these objects were genuine satellites.

3 SATELLITE STELLAR MASS FUNCTIONS

Our base COSMOS catalogue includes large numbers of intrinsically faint, low-redshift galaxies, and thus it provides a good opportunity to study the low-mass end of the SSMF. In this section, we will explore the SSMF and various related quantities, and their dependence on environment (i.e. primary halo mass). We will then compare our results to previous measurements from the literature.

Laigle et al. (2016) derived stellar mass estimates for the COSMOS2015 catalogue using a library of synthetic spectra generated by the stellar population synthesis model of Bruzual & Charlot (2003), the same method also described in Ilbert et al. (2015). They also estimated the stellar mass completeness limit empirically, based on the masses of detected galaxies and their K_s magnitudes relative to the limiting magnitude in this band. For the redshift range of 0 < z < 0.35, they estimate a 90  per cent completeness limit of M_* = 10^8.4 M_⊙ for quiescent galaxies and M_* = 10^8.1 M_⊙ for the star-forming galaxies. In paper I, examining number counts, we concluded that our local sample is complete down to at least M_* = 10^8.2 M_⊙ for the redshift range (z < 0.25). For very low redshift systems (z < 0.07), our sample appears to be complete down to as low as M_* = 10^7.2–10^7.5 M_⊙. For primary masses, these limits are amongst the deepest published, as discussed in Section 3.3.

3.1 Absolute stellar mass functions

In Figs 2 and 3, we show the (cumulative) SSMF, binned in six ranges of primary stellar mass and six ranges of primary halo mass, respectively. The mass functions are calculated using Method B described above. Only satellites within 1.5 R_vir in projection are included in the total.

We note that since these are cumulative plots, the error bars in different bins of secondary stellar mass are not independent; as an indication of the overall significance of the clustering signal, we can consider the measured abundances at any single value of M_*. Also, as explained in paper I, the halo-background decomposition used to fit the clustering signal and set the initial satellite probabilities assumes and fits a power-law dependence on halo mass; although the connection to the final measured satellite abundance is indirect, this may smooth slightly the variation of the SSMF with primary mass. From the results of paper I, fitting the clustering signal in each bin of primary mass independently might add an additional scatter of 20–30  per cent to the normalization of the SSMF, particularly at lower primary masses. This is generally less than the errors shown here, however, so the net effect would be minor.

Overall, the mass function appears Schechter-like, with a power-law dependence on secondary stellar mass and an exponential cut-off at the high-mass end. The amplitude of the SSMF also scales fairly smoothly with primary stellar or halo mass (given the caveat about smoothing discussed above). A similar conclusion was reached in paper I (see e.g. fig. 10). Previous work by Wang & White (2012) claimed a change in the slope of the (differential) SSMF with primary mass, but subsequent work (e.g. Zu & Mandelbaum 2015; Lan, Ménard & Mo 2016) suggests that this comes from fitting a single power law to a multicomponent SSMF that was sampled over different ranges of secondary mass for different ranges of primary mass. The true SSMF probably includes Schechter-function components for red and blue satellites at the faint end, plus a separate lognormal component of red galaxies above M_* ∼ 10^9.5–10¹⁰ M_⊙ (Lan et al. 2016). While there is a faint suggestion of this lognormal component in our 1–2 most massive primary ranges, we lack the SNR to distinguish it clearly. For less massive primaries, we clearly see the exponential cut-off in the SSMF at large masses.

Satellites in the lowest stellar mass bin on Figs 2 and 3 are approaching the stellar mass completeness limit of 10^8.2 M_⊙ mentioned above. This is particularly true for massive primaries (where the slope of the SSMF appears to flatten slightly below 10⁹ M_⊙), as these are rare, and tend to lie at higher redshift within our target volume (see Fig. A1 in Appendix  A); the highest mass bin, for instance, has no primaries below z  = 0.22. We suspect completeness effects towards the high-redshift limit of the sample may reduce the counts slightly at faint magnitudes.

3.2 Relative stellar mass functions

The subhalo mass function predicted by theory is approximately scale invariant, that is, the number of subhaloes with a given fraction of the main halo mass or circular velocity is roughly independent of primary halo mass (e.g. Moore et al. 1999). Alternately, since the cumulative subhalo mass function goes roughly as (M_sub)⁻¹, this scale invariance also implies that the number of subhaloes of a given (dark matter) mass M_sub per unit primary halo mass is roughly constant (e.g. Gao et al. 2004).

We can compare this prediction to the equivalent quantity in terms of stellar mass, that is, the abundance of satellites of a given stellar mass per unit primary halo mass. We refer to this as the net ‘efficiency’ of subhalo occupation on these scales, assuming a monotonic relation between subhaloes and satellites. Fig. 4 shows this efficiency or relative abundance, for the same bins of primary halo mass as in Fig. 3. At the low-mass end, we see that the SSMF per unit halo mass is constant to within the uncertainties, at least for the top four primary halo mass bins. At the high-mass end, the relative mass function is truncated when the satellite mass is roughly equal to the primary mass; this is a natural consequence of our requirement that |$M_*^{\rm sat}\lt M_*^{p}$|⁠.

Overall, these results are consistent with subhalo occupation being fairly universal at the masses probed here; that is, it seems that a subhalo of a given dark matter mass will tend to host a satellite galaxy of a set average stellar mass, independent of environment (where by ‘environment’, we mean primary halo mass). We can compare these results to those of Zu & Mandelbaum (2015). Down to primary halo masses of log (M_*/M_⊙) = 12.2, they measure a constant efficiency in the differential counts for satellite masses log (M_*/M_⊙) = 8.5–10.2. The value they measure does vary with primary mass at satellite masses log (M_*/M_⊙) ≳ 10.2. This will shift the cumulative mass functions slightly, but given the steep slope of the counts we might expect cumulative counts to agree to within ∼20  per cent for the top four bins shown in Fig. 4, which is certainly consistent with our results. Thus, our measured efficiencies are consistent with those of Zu & Mandelbaum (2015), over the primary mass range where they overlap.

At the lowest primary halo masses, we do see a possible indication of a drop in efficiency, but only at 1–2σ level. To test or demonstrate universal subhalo occupation conclusively, we would need higher SNR data, as well as more information on the radial distribution of satellites in different environments (particularly around low-mass primaries), to show that subhaloes of a given mass are occupied in exactly the same way.

Some evolutionary effects, such as infall due to dynamical friction, should depend on environment, however; specifically, they will depend on the subhalo-to-main halo mass ratio. Relatively massive satellites (i.e. small primary–secondary mass ratios) will experience stronger dynamical friction, and will merge faster, depleting the observed satellite population. We can look for evidence of such effects by measuring satellite abundance in terms of the corresponding stellar mass ratio μ_* ≡ M_{*, main}/M_{*, sat}. We will call the abundance as a function of this ratio the ‘relative stellar mass function’ (RSMF).

Fig. 5 shows the RSMF, estimated using Method B, for the same primary halo mass bins used previously. Note that since we have applied a fixed stellar mass completeness cut to the base catalogue at 10^8.2 M_⊙, the relative depth varies with the primary stellar mass. Points where satellites around more than half the primaries would lie below this completeness limit for a given stellar mass ratio are excluded. We can see that the top mass bin has a very good SNR, and probes the RSMF down to 3.4 dex in stellar mass ratio. For smaller haloes we only reach depths of 1.5–2 dex.

For the higher primary mass bins, the amplitude of the RSMF varies as expected from Fig. 4. For the lower bins, the overall scaling is harder to determine given the low SNR, but it seems to flatten, and may even be inverted. This could be a result of the flatter stellar-to-halo mass relation (SHMR – e.g. Leauthaud et al. 2012; Behroozi et al. 2013; Grossauer et al. 2015; Shuntov et al. in preparation) at these halo masses (which would produce more overlap between the primary halo mass bins), but it may also indicate residual background contamination problems in the lowest bins.

The shape of the RSMF is similar in all bins, with a fairly shallow slope at large mass ratios (low-mass satellites), and a steeper slope at small mass ratios (massive satellites). There is a slight indication that the rise at small mass ratios is steeper for less massive primaries than for more massive ones. Re-binning the results into two bins of primary mass to increase the SNR, we find that for higher primary masses (M_h > 12.8), d ln N/d ln μ_* ∼ 1.5 ± 1, whereas for (M_h < 12.8), d ln N/d ln μ_* ∼ 3 ± 2. Thus, there is a difference, but it is less than 1σ significance. Here again, this difference would be expected from the flatter slope of the SHMR at low masses. For a flatter SHMR, the range of μ_* covered by the first bin in the figure would map onto a smaller range of dark matter mass ratios. Thus, the first data point would represent a purer sample of major mergers, and dynamical friction effects would be stronger in this bin. Clearly, it would take a larger sample to confirm this. We revisit this point and show further evidence for dynamical friction effects below, however, in Section 5.

3.3 Comparison with literature results

The satellite luminosity function and stellar mass function have been measured many times, using different techniques to identify satellites individually or statistically. Fig. 6 shows one comparison of some previous measurements, in terms of their limiting secondary stellar mass sensitivity and primary halo mass coverage. Symbol types indicate measurements of the luminosity function (triangles), the stellar mass function (squares), both (pentagons), or the relative luminosity function (the number as a function of magnitude offset – circles). Note that the limits are approximate, as they have been converted from various bands and/or assume different model mass-to-light ratios and primary SHMRs. Different surveys also vary enormously in size and SNR. With these caveats, we see that our current work constrains satellite abundance at some of the lowest primary halo masses probed to date. The sensitivity in secondary mass is less exceptional, although at very low redshift we do in principle have some sensitivity down to masses of log (M_{*, sat}/M_⊙) ∼ 7.5 (not shown here). The only comparable work is that of Sales et al. (2013); they obtain a positive detection of the satellite population around hosts with nominal stellar masses log M_* = 7.5–8, down to a magnitude difference of 1.4, which we translate to an approximate limiting stellar mass of log M_* = 7.2 for the satellites.

Given these previous results, we will compare the measured amplitude of the SSMF to the values in Wang & White (2012) and Lan et al. (2016), since these are the deepest published SSMFs; for the shape of the SSMF, we will convert the relative luminosity function published by Mao et al. (2021) to stellar mass.

Fig. 7 shows our measured amplitude of the SSMF, as a function of primary halo mass, compared to the results of (Wang & White 2012, – blue) and (Lan et al. 2016, – cyan). We have used the value of the cumulative mass function at M_{*, sat} = 10⁹ M_⊙, as a measure of the amplitude, as this point is reasonably well sampled over the whole range of primary halo mass considered. Symbols show the halo mass range in each bin as horizontal width, and the uncertainty as height (the uncertainties for Wang & White (2012) and Lan et al. (2016) were not directly reported, but are assumed to be small – thus the nominal height shown here). Note that the different studies also used slightly different limiting radii to define the satellite abundance – Wang & White (2012) used a fixed radius of 300 kpc (or 170 kpc in their lowest mass bins), Lan et al. (2016) used the virial radius R_vir, which should have a value of ∼250 kpc for primary halo mass log (M_{h, p}/M_⊙) = 12, while we have used R_vir ∼ 375 kpc at log (M_{h, p}/M_⊙) = 12. Testing our model with different radial cuts, we expect these differences may shift the measured amplitude by ∼20  per cent between Lan et al. (2016) and Wang & White (2012), and a further ∼20  per cent between Wang & White (2012) and ourselves. These shifts would slightly improve the agreement between our results and Wang & White (2012), particularly at the high-mass end, but they are relatively small compared to the uncertainties, so we have not included them here.

Overall, we see excellent consistency between our measured amplitude of the SSMF and the values found in previous work; while there appears to be a slight offset between results around log (M_{*, p}/M_⊙) = 10.8, it is well within our uncertainties. The amplitude appears to vary as a simple power law in log M_{*, p}, over four decades in primary stellar mass.

We can also test the shape of the SSMF, or equivalently the RSMF, at a given primary mass. Here we compare to Mao et al. (2021), who recently presented the data release from the SAGA II spectroscopic survey, as well as some initial science results, including a relative luminosity function. Estimating stellar masses for the SAGA sample as described in Appendix  B, we can use their satellite and host catalogues to construct an RSMF for MW-like hosts.

The comparison is shown in Fig. 8. Once again, the agreement is excellent; over the range of mass ratios log (M_{*, p}/M_{*, s}) = 0.5–2, the SAGA results lie well within the uncertainties of our RSMF measurements. Our results appear to roll over, becoming incomplete around mass ratios of μ_* = 2–2.5, depending on the primary mass bin; the SAGA results extend deeper, with rising counts down to a mass ratio of around μ_* = 3. (Given the SAGA II magnitude limit is about M_r = −12.3, and their hosts are selected in the range −23 > M_K > −24.6, which corresponds approximately to −20.5 > M_r > −22.1, the SAGA results should be relatively complete down to the last few points plotted, or μ_* ∼ 3.2.)

We note that Mao et al. (2021) include only satellites at separations R_p < 300 kpc and velocity offests Δv < 250 km s⁻¹, which are slightly more restrictive than our limits. Judging from fig. 8 of their first paper (Geha et al. 2017), this might reduce their measured satellite abundance by ∼30  per cent relative to ours, but this is still consistent with our results (and in fact a slightly better match at low mass ratios/large satellite masses).

Overall, we conclude that for the ranges of primary and secondary mass where we can directly compare them, both the normalization and shape of our SSMF (and/or RSMF) are in excellent agreement with previous results. The clustering method has allowed us to extend these previous measurements to very low primary masses, however, despite the relatively small COSMOS field, and thus shows great potential for future surveys.

4 SATELLITE-CENTRAL CONNECTION

Given the overall consistency of our SSMF measurements with previous results, we will proceed to consider satellite populations as a function of primary properties. In particular, there is already considerable evidence that primary morphology influences the abundance and properties of satellites:

• Several previous studies find that red/blue primaries have a higher/lower abundance of satellites at fixed stellar mass (e.g. Wang & White 2012; Mandelbaum et al. 2016).

• Central galaxies of different morphological types may follow different SHMRs (Wojtak & Mamon 2013; Hudson et al. 2015; Mandelbaum et al. 2016; Correa & Schaye 2020, Spitzer et al. in preparation), which might or might not explain the first point.

• Satellite populations also show ‘conformity’ in colour, that is red primaries tend to have more red satellites, while blue primaries have more blue satellites (e.g. Weinmann et al. 2006; Wang & White 2012; Hartley et al. 2015; Knobel et al. 2015).

These patterns could reflect assembly bias (Gao, Springel & White 2005), red primary galaxies lying preferentially in denser regions where structure has formed earlier and dwarf galaxies are more abundant, but satellites tend to be quenched. Alternately, they could be due to effects at the single-halo scale, including a different SHMR for central galaxies with different colours or morphologies. If central morphology depends on the detailed merger history of the system, this may also influence the satellite population; a simple example is fossil groups (Ponman et al. 1994), where recent mergers appear to have depleted the bright end of the satellite luminosity function.

To test for these effects, we consider satellite populations around subsamples of primaries split by colour or by sSFR, compare our results to previous measurements from the literature, and also consider the question of conformity.

4.1 Satellite populations split by central colour or sSFR

First, we consider the division of primaries into subsamples. For colour, we choose to split on the index c = (B − i⁺), as explained in Appendix  C. We use a colour cut at c = 1.6 for massive primaries (log (M_*/M_⊙) > 10), while for less massive primaries, we move this to c = 1.5, to reflect the evolution of the red sequence at lower stellar mass. For sSFR, we split at the value −11, which produces roughly comparable subsamples of star-forming and passive galaxies, as explained in Appendix  C.

Fig. 9 shows the SSMF in subsamples split by colour. Note that the signal of the highest primary mass bin is entirely from red primaries, as the primary sample has no blue galaxies in this mass range. While our results are consistent with those mentioned above (the SSMF for red primaries lies above the SSMF for blue primaries), we lack the SNR to reach significant conclusions. In the intermediate primary mass bin, there is a significant detection of massive satellites (log (M_{*, sat}/M_⊙) > 10) around red primaries, versus no detection around blue primaries; thus, the two populations differ at the ∼1.5σ level. (This measured difference is also consistent with earlier results, e.g. Wang & White 2012.)

A number of previous studies have indicated that the host haloes of red primaries are, on average, more massive than those of blue ones (e.g. Wojtak & Mamon 2013; Mandelbaum et al. 2016; Correa & Schaye 2020). Correa & Schaye (2020), for instance, find that disc galaxies have stellar masses up to 1.5 times larger at fixed halo mass, at the high-mass end of the galaxy population. Given the slope of the SHMR, this is equivalent to a factor of ∼2 difference in halo mass at fixed stellar mass. Earlier work by Mandelbaum et al. (2016) found a similar result: for primary stellar masses log (M_*/M_⊙) = 10.3–11.6, the haloes of passive centrals are at least twice as massive as those of star-forming centrals of the same stellar mass. If we assume that total satellite number is directly proportional to host halo mass (as suggested by Fig. 4), we would expect a higher number of satellites around red primaries than around blue primaries of equal stellar mass. The amplitude of this effect is only a factor of ∼2, however, which is within the uncertainties in Fig. 9. There could also be other factors that affect the normalization of the SSMF; if the satellites of red or passive primaries are more often quenched themselves (‘conformity’), they may have a different subhalo SHMR, which could cancel out some or all of the effect of the primary halo mass difference. We consider conformity below and quenching rates in Section 5.

We have also measured the SSMF in subsamples split by sSFR, as shown in Fig. 10, to test whether they showed more evidence of systematic differences in satellite populations. At the highest mass bins, we find no significant difference between the satellite abundance around high-sSFR primaries and low-sSFR ones. At intermediate and low primary masses, low-sSFR primaries seem to have more satellites, consistent with previous results, but once again the SNR of the detection is only about 1σ.

4.2 Conformity in the spectroscopic sample

While we lack the SNR to properly test conformity with the clustering method (our attempts to measure it were inconclusive), the spectroscopic sample does show some evidence for this effect. First, Fig. 11 shows the distribution of colour versus stellar mass for the spectroscopically confirmed secondaries (a similar plot is shown for the primaries in Appendix  C). We see a strong correlation between the stellar mass and colour, with a well-defined red sequence visible at the high-mass end.

Fig. 12 then compares the colour indices of primary and secondary galaxies. The horizontal line indicates the division between red and blue primaries (at high primary mass), while the horizontal line indicates the median colour of the secondary sample. Examining each quadrant, we see that red secondaries are rare around blue primaries. Thus, red satellites occur mainly around red primaries, while blue satellites occur around a range of primaries; also the satellites of blue primaries tend to be blue, while the satellites of red primaries have a range of colours. While this pattern is broadly consistent with previous measurements of conformity (e.g. Weinmann et al. 2006; Wang & White 2012; Hartley et al. 2015; Knobel et al. 2015), we caution that the spectroscopic sample is inhomogeneous and incomplete, particularly for the more massive primaries, which tend to lie at higher redshift (see Appendix  A). Thus, it is possible that we are missing some faint blue satellites around distant massive red primaries (although it seems less likely that we are missing red satellites around nearby low-mass blue primaries).

5 ENVIRONMENTAL QUENCHING

The quenching of star formation in a galaxy can be driven by internal processes (e.g. rapid gas loss via violent starbursts) or environmental ones (e.g. ram pressure stripping by the intra-cluster medium). There are clear indications of environmental quenching even on small scales; in the Local Group, for instance, most nearby dwarfs are quiescent, and star-forming dwarfs are generally distant, with the notable exception of the Magellanic clouds (McConnachie 2012). Studies of environmental quenching consider either the total quiescent fraction in the satellite population or the ‘environmentally quenched fraction’. This is the fraction of those galaxies that would be expected to be star-forming in the field, but that are observed to be quiescent in satellite populations. It can be calculated as

where f_s refers to the quiescent fraction of satellites and f_f refers to the quiescent fraction of field galaxies. Thus, for instance, if f_f  = 0.2 and f_s = 0.8, we conclude that the fraction of (star-forming) systems quenched by their environment is 0.6/0.8  = 75  per cent.

The environmentally quenched fraction has previously been found to depend on secondary stellar mass, with hints of an abrupt change around a secondary stellar mass log (M_*/M_⊙) ∼ 8 (Wheeler et al. 2014), but measurements in the LG (Wetzel et al. 2015) and in the SAGA II survey (Mao et al. 2021) do not confirm this feature, and in general there is limited information in the secondary mass range 10⁷–10⁹M_⊙.¹ Quenching may also vary as a function of primary properties, as mentioned above, so we will consider it in the various primary subsamples defined in Section 4.

To distinguish quiescent galaxies from star-forming ones, we use the ‘CLASS’ flag in the COSMOS2015 catalogue, which classifies galaxies based on their location in the (NUV − r)–(r − J) colour–colour plane (Laigle et al. 2016). While reviewing this classification, we noticed, however, that about 30  per cent of the galaxies in the base catalogue do not have valid NUV-band measurements. In principle, these galaxies should not have been classified due to the lack of NUV − r colour. In practice, they were classified as ‘CLASS = 1’ (or star-forming) in the COSMOS2015 catalogue. To correct this, we apply our own classification, similar to that of Laigle et al. (2016), but replacing the NUV − r colour with ‘MNUV_MR’, an estimated colour given in the catalogue that appears to be available for most objects, and is also corrected for dust extinction. For galaxies where valid NUV measurements are available, we have compared the two colours. Because it is extinction corrected, ‘MNUV_MR’ is slightly bluer than the colour NUV − r for star-forming galaxies. For quiescent galaxies, however (that is galaxies with MNUV_MR > 3.1), it is very similar, with a mean difference of −0.05 and a r.m.s. scatter of ∼0.175 magnitudes. Using the MNUV_MR colour ensures complete classification for all of the objects in our base catalogue.

The new classification ‘CLASS_XI’ is defined as:

• ‘CLASS_XI=0’ (Quiescent) if [‘CLASS = 0’ OR (‘MNUV_MR>3.1’ AND ‘MNUV_MR>3 × (r − J) + 1’)];

• ’CLASS_XI’ = 1 (Star-forming) for the rest.

5.1 Quenching versus secondary stellar mass

To estimate the environmentally quenched satellite fraction, we can proceed in two ways. First, we can consider the whole catalogue (binned by stellar mass) and use the previously calculated overall satellite/field probabilities, P_sat and P_field, without reference to particular primaries.

Suppose the catalogue contains N_{q, tot} quiescent galaxies in a given stellar mass bin. We can divide this into satellite and field or primary populations, such that

On the other hand, the total number of satellites can be written:

and we can estimate the number of quiescent field galaxies by assuming an universal quiescent fraction for field galaxies within the given mass bin, f_{q, field}, such that

Combining these expressions, the quiescent satellite fraction f_{q, sat} is

To estimate f_{q, field}, we select objects with P_field > 0.99 as the field galaxy subset, and measure the field quiescent fraction as a function of stellar mass, based on this subset. For the satellite sample, including galaxies with very low P_sat will lower the SNR of satellite quiescent fraction, but restricting the sample to objects with the highest values of P_sat will limit the sample size. As a compromise, we choose to limit the sample to objects with P_sat > 0.8 for the calculation in equation (7).

Fig. 13 compares the quiescent fractions as a function of stellar mass for these different galaxy populations. In the legend, ‘Total’ (filled black circles) indicates the whole base catalogue. ‘Field/Central’ (blue squares) indicates objects likely to be field galaxies, defined as above, while ‘Satellites’ (red triangles) indicates objects likely to be satellites. The quiescent fraction for the satellites is estimated using equation (7).

Overall, the quiescent fraction of all galaxies in the base catalogue is low, but increases both towards the high-mass end and towards the low-mass end. The trend at the high-mass end is expected as a result of the increasing mass quenching effect for high-mass field galaxies. We suspect the increase at low masses is largely contributed by satellites. Note that we do also see a small increasing trend at the low-mass end for field galaxies as well, which may come with an increasing fraction of mis-classification between field galaxies and satellites at very low masses. The field galaxies have significantly lower quiescent fractions than the average of all galaxies, implying a high quiescent fraction for satellites. The significant difference between the satellites and field galaxies provides strong evidence for environmental quenching effects.

Note that this discussion assumes the COSMOS2015 definition of quiescence; if we consider instead the red fraction with c > 1.5 or 1.6, as defined in Section 4.1, we find red fractions of 33  per cent,  54  per cent, and 93  per cent (or 22.5/54/93  per cent for a uniform cut at c = 1.6) for the top three stellar mass bins shown in Fig. 13. These are ∼20  per cent higher than the red fractions measured for isolated samples (e.g. Geha et al. 2012; Wang & White 2012), but closer to the fractions measured in Wang & White (2012) for the SDSS main sample. The difference may be partly due to the colour index we use, but it probably reflects the fact that the isolation cuts in these studies were quite strict, whereas our field probability calculation includes some primaries in denser regions.

For satellites, the quiescent fraction is fairly high, around 0.4–0.9. This is comparable to the result of Wetzel et al. (2015) but systematically higher than the fraction reported in Mao et al. (2021) except for the lowest stellar mass bin around 10^7.5M_⊙. Recently, Carlsten et al. (2022) also found a higher quiescent fraction than Mao et al. (2021); their results vary a bit with sample selection but generally lie in the bottom third of the (orange) uncertainty range given for Wetzel et al. (2015). (Note that they determine quiescence from UV and optical colours, similarly to our method.) They suggest that the SAGA catalogue may be incomplete for low surface brightness passive dwarfs, but also show that more generally, the quiescent fraction depends on primary stellar mass, as discussed below. This dependence could explain much of the variation between the different reported results.

Above log (M_*/M_⊙) = 10, the measured fractions are very uncertain, as we have very few pairs in this mass range with satellite probabilities of 0.8 or more. Below log (M_*/M_⊙) = 8, the quiescent fraction drops significantly, likely due to a higher fraction of misclassified field/satellites at the low-mass end. (Note also the comment in Appendix  D about possible bias in the estimated quiescent fraction at low masses.)

5.2 Quiescent fraction versus primary morphology and stellar mass

As noted earlier, quenching may be correlated with central properties, so we will also consider the satellite quiescent fraction for the red and blue primary subsamples defined in Section 4, as well as for several bins of primary stellar mass. To calculate these fractions, we follow the procedure described above (equation 7), but first split the satellite samples by primary properties, while still using the whole sample to estimate f_{q, field}. For instance, we can calculate the satellite quiescent fraction around red primaries as

where the notation ‘RP’ indicates quantities measured around red primaries. A single galaxy will often be a potential satellite of several primaries. In these cases, we associate the satellite with the central for which it has the largest satellite probability, although this may introduce some noise into our split by primary properties, by mixing satellites from the two subsamples.

Fig. 14 shows the quiescent fraction for red and blue primary subsamples, as well as for the entire base catalogue. We note that the SNR of the blue sample is lower, due to a smaller sample size. Despite the large uncertainties, we can see that the satellites around blue primaries show a significantly lower quiescent fraction than the ones around red primaries, over the stellar mass range of log (M_*/M_⊙) = 8.5–10. We note that a difference in the quiescent fraction of satellites around red and blue primaries might affect the normalization of the SSMF, as discussed above. The contribution of the high satellite quiescent fraction of red primaries may also explain why our measurements are higher than the MW results from Wetzel et al. (2015) at the stellar mass of log (M_*/M_⊙) ∼ 9. Our results for the blue primaries are in excellent agreement with the recent measurements by Carlsten et al. (2022), who find quiescent fractions of ∼0.7/0.4/0.2 at log M_* = 7/8/9 for their sample of (mainly star-forming) primaries.

Finally, in Fig. 15, we test the dependence of the quiescent fraction on primary stellar mass. For the mass bins plotted, low-mass primaries generally have lower quiescent fractions, but the SNR is too poor to reach a definitive conclusion. If we bin together the results for the secondary mass range log (M_*/M_⊙) = 7.5–10, we find fractions 0.64 ± 0.04 for the higher primary mass range, versus 0.52 ± 0.15 for the intermediate mass range, a difference of ∼0.8σ. For the lowest mass range, there is too little SNR to reach any conclusion. The recent results of Carlsten et al. (2022) confirm the dependence on primary stellar mass, showing the quenched fraction increases by ∼50  per cent over the K_s-band luminosity range −22.5 to −25. The fact that the quiescent fraction may be lower around low-mass primaries motivates us to consider its dependence on the primary-to-secondary mass ratio as well; we will discuss this next.

5.3 Quiescent fraction versus relative mass

Dynamical friction will drag any satellites that are comparable in mass to the primary into the centre of the main halo on a very short time-scale, equivalent to a few orbits (e.g. Colpi, Mayer & Governato 1999; Taylor & Babul 2004). By implication, the most massive surviving satellites are necessarily recent mergers. We anticipate that this could produce a selection effect, whereby satellites with large relative masses are still star-forming, and the quiescent fraction depends on the primary-to-secondary mass ratio.

Colpi et al. (1999) have estimated the time-scale for infall due to dynamical friction, on an orbit of a given initial energy and angular momentum, as

where J_circ and r_circ are the angular momentum and radius of a circular orbit of the same energy, M_{h, m} and M_{h, s} are the mass of the primary halo and the mass of the satellite halo at the initial time, μ = M_{h, m}/M_{h, s} is the ratio of the two, and ϵ = J/J_circ is the initial circularity of the orbit (e in the equation is simply Euler’s number). The second equality assumes the energy of the orbit is equal to that of a circular orbit at the virial radius (with period P_vir), while in the final equality, we have substituted the radial orbital period P_rad (Binney & Tremaine 2008). By comparison, the first and second pericentric passages, where tidal effects are strongest and quenching through triggered starbursts is likely to occur, take place after approximately 0.2 P_rad and 1.2 P_rad, respectively.

Suppose satellites are all star-forming on initial infall into the main system (as indicated by Fig. 13). If systems are completely quenched at the first pericentric passage, and then fall into the centre of the main halo and merge by τ_DF, then assuming a uniform distribution of infall times, the quiescent fraction of the surviving satellites will be (τ − 0.2)/τ = 1 − 0.2/τ, where τ ≡ T_DF/P_rad. Similarly, if systems are only quenched after the second pericentric passage, the quiescent fraction should be 1 − 1.2/τ.

Fig.16 shows the quiescent fraction for our satellite sample, binned by mass ratio (points with error bars). We see a clear difference between the first data point and the subsequent ones; it is ∼2σ below the next bin, and deviates from the average of the others by almost 3σ. The smooth curves indicate the expected fraction as a function of mass ratio, if quenching is 100 per cent efficient at the first pericentric passage (short dashed lines), 100 per cent efficient at the second pericentric passage (solid lines), or 35 per cent efficient at the first and 35 per cent efficient at the second (long-dashed line) . In the first two cases, the three curves are for orbital circularities ϵ = 0.9, 0.5, 0.1 from top to bottom; in the final case, for clarity we show only the results for ϵ = 0.5. While we have not fit the data explicitly to any of these models, a model with partial quenching at each of the first two pericentric passages clearly matches the general trend in our results.

In the limit of large mass ratios, the quiescent fraction goes to ∼0.65, rather than 100  per cent. This could be because a significant fraction of all satellites have fallen in recently (e.g. Taylor & Babul 2005, suggest the median number of orbits spent in the main halo is around 2 for low-mass satellites); it could be because some satellites are on circular orbits with large pericentres, and don’t experience strong tidal triggering, or it could be because quenching/triggering is not 100  per cent effective at quenching galaxies, even on radial orbits. We also note that the lowest bin may be affected by misclassification of near-equal-mass secondaries. Since the scatter in the SHMR is ∼0.16 dex (cf. Paper I), the uncertainty in the primary-secondary mass ratio is approximately 0.25 dex, so a +1σ deviation could result in an primary and secondary with the median mass ratio of the first bin being switched. Assuming this positive deviation happens 16  per cent of the time, and that the field quenched fraction is essentially zero (cf. Fig. 13), this will reduce the quiescent fraction measured in this bin by 16  per cent. The red diamond shows the value after correcting for this effect.

5.4 Quenching: summary

We can summarize our results on environmental quenching as follows:

• The quiescent fraction in our satellite populations is significantly higher than that in the field populations, or the base sample as a whole. This confirms that the clustering method (which is independent of colour or star-formation status) is correctly identifying a distinct population.

• The average fraction is comparable to that for the Local Group or even higher, which may reflect the presence of fairly massive primaries in our sample.

• The quiescent fraction in our satellite populations does not appear to depend strongly on secondary mass (Fig. 13), at least not over the range of secondary stellar masses log (M_*/M_⊙) ∼ 8–10 where we can measure it reliably.

• The quiescent fraction may depend on primary colour (cf. Fig. 14); blue primaries appear to have lower quiescent fractions then red primaries, though the difference is only significant at 1σ. This is consistent the idea of ‘conformity’, for which there is clear evidence in the spectroscopic sample (cf. Section 4.2).

• The quiescent fraction may depend on primary mass, with a higher fraction in more massive haloes (cf. Fig. 15). Though we lack the SNR to establish this conclusively here, it is consistent with other recent results in the literature.

• There is stronger evidence for the quiescent fraction depending on the primary-to-secondary mass ratio (cf. Fig. 16). This is consistent with a model where the quiescent fraction reflects the mean accretion time of satellites, and massive satellites that have only recently fallen in to the main halo are not yet quenched (see Section 5.3).

Given the complex, multivariate dependence of quiescence on primary and secondary properties, we clearly need more data to map out these trends in detail. For low-mass groups, dedicated spectroscopic campaigns targeting local examples, such as Mao et al. (2021) or Carlsten et al. (2022), are probably the most promising approach, as they provide additional information about satellite orbits, quenching times and infall times.

6 SUMMARY AND OUTLOOK

In paper I, we established a new, iterative method for quantifying satellite abundance using clustering in catalogues with accurate photo-zs. The method allows us to measure the clustering signal even in crowded fields, avoiding biases that may arise from selecting only the most isolated systems. We tested the method using the COSMOS2015 photo-z catalogue of Laigle et al. (2016), which has excellent photo-z accuracy, but covers only a very small field.

In this paper, we have explored the properties of the satellite populations detected in the COSMOS field. We measure the amplitude and shape of the SSMF, and find results for both that are consistent with previous measurements by Wang & White (2012) and Lan et al. (2016), but extend these down to primary masses log (M_*/M_⊙) ∼ 10.2. We also measure the SSMF per unit halo mass, an indicator of the net efficiency of galaxy formation in different environments, and find fairly constant efficiency at primary halo masses log (M_h/M_⊙) ≳ 12, consistent with previous studies (e.g. Zu & Mandelbaum 2015), while at lower primary halo masses there is some marginal evidence for reduced efficiency. Expressing the SSMF as a function of the primary-to-secondary mass ratio (the ‘RSMF’), we find marginal evidence for dynamical friction effects depleting the low-mass ratio end of the satellite population (i.e. relatively massive satellites). Splitting the primary sample by colour, we find some evidence for greater satellite abundance around red primaries, particularly for large secondary masses. Here again, this is consistent with previous results where we overlap in primary mass (e.g. Wang & White 2012). Examining the spectroscopic subsample within our data, we also see evidence for conformity in colour; in particular, red satellites appear rarer around blue galaxies. Finally, we study quenching as a function of secondary mass and primary properties, seeing evidence for a number of trends. The most significant one (and previously unreported, to our knowledge), is a selection effect whereby the satellites with the largest relative masses have lower quiescent fractions (cf. Fig. 16), because they have only merged into the main halo recently, and have not yet been quenched by pericentric passages.

Overall, our results show good consistency with previous studies, and extend these to lower primary mass; they are limited, however, by low SNRs and the small size of the COSMOS field. Many large surveys currently underway, or planned for the near future, will greatly increase the deep imaging data available; on the other hand, most will also have much larger photo-z errors, relative to COSMOS. One of the most promising surveys for applying our method is the deep polar-cap component of SPHEREx², an all-sky survey satellite with a wide-field spectral imager that will produce low-resolution (R∼20–100) spectra, with a final redshift accuracy similar to COSMOS, but over an area 50 times larger (see Paper I). The resulting increase in the SNR of the clustering signal should confirm or rule out many of the marginal trends seen in the current work, and further clarify the complex relationship between satellite and central galaxies.

ACKNOWLEDGEMENTS

We thank N. Afshordi, A. Broderick, M. Balogh, G. Geshnizjani, M. Hudson, A. Kempf, M. Sawicki, and our friends and collaborators from the COSMOS survey for their comments and advice. JET acknowledges support from the Natural Sciences and Engineering Research Council of Canada, through a Discovery Grant. The COSMOS2015 catalogue is based on data products from observations made with European Southern Observatory (ESO) Telescopes at the La Silla Paranal Observatory under ESO programme ID 179. A-2005, and on data products produced by TERAPIX and the Cambridge Astronomy Survey Unit on behalf of the UltraVISTA consortium.

DATA AVAILABILITY

Most of the basic data presented in this paper are publicly available. The COSMOS2015 catalogue (Laigle et al. 2016) can be accessed from the COSMOS website, at http://cosmos.astro.caltech.edu/page/photom. A few spectroscopic redshifts that are unpublished from the COSMOS collaboration (Salvato, private communication) will be shared on reasonable request to the corresponding author with permission of the COSMOS collaboration. The derived data generated for this paper will also be shared on reasonable request to the corresponding author.

Footnotes

References

APPENDIX A: REDSHIFT DISTRIBUTION FOR THE SIX PRIMARY MASS BINS

The redshift distributions of our first-run primaries, in six different bins of halo mass, are shown in Fig.  A1. Dashed vertical lines indicate the median redshift for each bin. The median redshift clearly increases with primary halo mass, as discussed in the main text.

APPENDIX B: COMPARISON TO SAGA II (MAO ET AL. 2021)

To construct the RSMF for the SAGA II survey, we need stellar mass estimates for both primaries and secondaries. A full set of primary stellar masses were not published in Mao et al. (2021), though masses are given for a subset of the same sample in Geha et al. (2017). We estimate the remaining masses from the published K-band absolute magnitudes, using a mean mass-to-light ratio derived from the published values:

where C_S1 = 1.042 is the median of log [M_*/M_⊙] + 0.4M_K for the subsample in Geha et al. (2017).

The satellite stellar masses published in Mao et al. (2021) were instead estimated by using the r-band magnitude and g − r colour:

where M_r, o refers to K-corrected r-band absolute magnitude and (g − r)_o is the K-corrected g − r colour.

APPENDIX C: PRIMARY SUBSAMPLES IN COLOUR AND SSFR

To separate red and blue primaries, we use c = B − i⁺ as our colour index. Among the various photometric bands for the COSMOS2015 catalogue, B-band is the deepest Subaru broad band that covers the whole field. For the other band, we select i⁺ for the best combination of wavelength baseline, depth, and noise properties (cf. Laigle et al. 2016, table 3). For consistency, we use the magnitudes measured using a fixed 3 arcsec aperture in each case. We also applied K-corrections to our colour indices, using the public tool available at this URL³, which is based on the method in Chilingarian, Melchior & Zolotukhin (2010). As our filters are not included in this tool, we assume the nearest equivalents in colour; since our objects are fairly low redshift, the error introduced by the conversion should be minor.

Fig. C1 shows primary colour versus stellar mass. There is a clear correlation, as well as clustering of points at c = B − i⁺ > 1.6 that we identify as the red sequence. We will use this colour cut to define the red primary sample for log (M_*/M_⊙) > 10. For lower stellar masses, the red sequence moves bluewards due to metallicity effects, so we move our cut to c = 1.5. Note that by these definitions, all primaries with a stellar mass over log (M_*/M_⊙) = 10.9 are red.

The SFR and sSFR in the COSMOS2015 catalogue were derived along with the redshift and stellar mass, by using the best-fit templates from Polletta et al. (2007) and Bruzual & Charlot (2003). The catalogue includes median best-fit estimates, as well as upper and lower limits. Fig. C2 shows the sSFR versus colour for the first-run primary sample. To define a cut in sSFR that roughly corresponds to our colour selection, we choose the median value for the red sample, which is approximately sSFR = 10⁻¹¹yr⁻¹.

APPENDIX D: COMPLETENESS

D1 Photometric and redshift completeness

The COSMOS2015 catalogue (version 1.1) is described in (Laigle et al. 2016), and contains a total of 1.18 million objects. The nominal i⁺ limiting magnitude is m_AB =26.2/26.9 for 3σ detections in a 3 arcsec/2 arcsec aperture respectively. The catalogue was not selected from the i⁺ images, however, but from a χ² sum of the (Ultra-Vista) NIR and (Subaru) z⁺⁺ images. Comparison with previous i⁺-selected COSMOS catalogues shows that at most 3.5 per cent of galaxies brighter than i⁺  = 25.5 could be missing due to this selection in other bands, although the actual fraction is probably smaller, as some of these previous detections may correspond to fake sources or noise. (A comparison with a deeper NIR survey also indicates that the catalogue appears to be complete in NIR, down to its nominal depth.)

Of all the objects in the catalogue, approximately 79 per cent (933K) have valid i⁺magnitudes. Of the rest, most (20 per cent of the total) are located at the edges of the field, where the i⁺ coverage stops. For central part of field, only ∼1 per cent of catalogue are missing i⁺magnitudes, due to masking. The i⁺ depth does vary slightly across field (cf. fig. 6 of Laigle et al. 2016), but is always deeper than out limit 25.5. Samples taken around i⁺ = 25 show no large-scale gradients across the field. Of the objects with good i⁺ photometry, approximately 60 per cent (538K) are in the range i⁺< 25.5, and 98.9 per cent of these have good photometric redshifts.

Thus, conservatively, we conclude that our initial sample cut at i⁺ = 25.5 could be missing a few per cent of all galaxies in that magnitude range due to the catalogue selection in other bands, 1 per cent due to masking, and ∼1 per cent due to missing photo-zs, for a total incompleteness of ∼5 per cent. The true incompleteness in the satellite counts is probably less than this, since some of the i⁺-selected sources detected in previous catalogues may have been fake, and since our clustering method corrects for masking explicitly.

D2 Completeness in derived quantities

In our analysis, we also select subsamples using several other derived quantities, however, including stellar mass, B − i⁺ colour, and quiescent status (the CLASS flag), which in turn is based on NUV − r and r − J colours. Our base sample is also selected using cuts in redshift and the redshift errors. This raises two questions. First, do the galaxies in our base sample have reasonable values of these derived quantities, or does our base sample include objects with missing or unphysical values? And second, could the selection in redshift and redshift error bias the sample, e.g. by removing red/blue or active/passive objects preferentially?

Fig. D1 shows the magnitude distribution for the initial catalogue (after selection for valid i⁺ magnitudes), as well as for objects with missing or invalid B − i⁺ colours, stellar masses or sSFRs. We see that down to our magnitude limit of i⁺ = 25.5, only a few per cent of objects have missing or invalid colours, and almost all objects have valid stellar masses and sSFRs. Thus incompleteness in these quantities is not a significant problem.

In contrast to this, our analysis of the CLASS flag values in the COSMOS2015 catalogue found that these were often incorrect for low-redshift quiescent galaxies, as described in Section 5, due to missing or incorrect NUV magnitudes. After correcting these values using the |$MNUV\_MR$| colour in the catalogue, we find that all but a few per cent of the sample can be assigned a corrected quiescence classification ‘CLASS_XI’.

To address the second question, in Fig. D2 we show the photometric redshift error distribution for our base sample, as a function of z_pdf. The three panels show, from left to right, then entire sample, blue (B − i⁺ < 1.5), and red (B − i⁺ ≥ 1.5) galaxies. We see that our cut in redshift error σ_z is well away from the main part of the distribution, both for the whole sample and for the blue and red samples individually. The blue and red distributions are similar, although the former subsample is larger and thus the points sample the distribution more densely. Overall, our cuts should include 90 per cent or more of all objects with z_pdf in our target redshift range (z = 0–0.25), without any major bias against particular galaxy types. The excluded objects have very large redshift errors, so only a fraction of them will be genuine low-redshift galaxies. Overall, we estimate that any incompleteness due to the redshift and redshift error cuts is less than 5 per cent.

Combining this result with the basic photometric/redshift incompleteness derived above, we estimate an upper limit of 10 per cent on the total fraction of objects in our nominal selection range (i.e. galaxies with magnitudes i⁺ < 25.5 at redshifts z < 0.25) that could be missing from our sample, and 1–2 per cent on the fraction of objects in our sample with missing or incorrect values of the derived quantities used to define subsamples in our analysis. We conclude that incompleteness effects are generally much smaller than the statistical errors on our results. There is one additional source of error or bias that is not completely negligible, however; we discuss this below.

D3 Redshift error bias in the quenched fraction

At low redshift, the photo-z errors of faint blue/star-forming galaxies are generally slightly larger than those of red/quiescent galaxies, as suggested by the middle and right panels of Fig. D2. Below log M_* = 8.5, the mean error for the former subset is 20–30 per cent higher than for the latter. Since the depth of the ROI around each primary scales with the redshift error of the secondary (through the uncertainty in the velocity difference, Δv), this means that blue galaxies will sample a slightly larger ROI than red galaxies, and more of them will be included in the background estimate. Thus, the blue-to-red fraction in the field will be overestimated slightly, and, after subtraction, the blue-to-red satellite fraction will be underestimated, or equivalently the quiescent fraction in the satellite population will be overestimated.

While it is difficult to correct for this effect exactly without modelling the redshift error distribution in more detail, we can estimate the magnitude of the effect by adding an extra 1σ scatter to all redshifts, and seeing how the quiescent fraction varies as a function of secondary stellar mass (as in Fig. 13). We find that the resulting changes in quiescent fraction are less than 0.5σ for all except the lowest mass bin, which moves down by 1σ. Thus, while this bias is not completely negligible and should be reconsidered in future work, it does not dominate our statistical errors.