Main-sequence Scatter is Real: The Joint Dependence of Galaxy Clustering on Star Formation and Stellar Mass

We use galaxy redshifts from Data Release 10 of the Sloan Digital Sky Survey (SDSS; Ahn et al. 2014). Stellar mass and SFR measurements are taken from the MPA-JHU catalog (Kauffmann et al. 2003; Brinchmann et al. 2004). In this catalog, fiber SFRs are measured from Hα (for star-forming galaxies) and estimated from the D4000 break (for quiescent galaxies) for the light within the SDSS fiber. Light outside each galaxy's fiber is converted to an SFR assuming the same average SFR/luminosity ratio as other fibers with similar g − r and r − i colors. The total SFRs used here are the sum of the fiber and nonfiber SFRs.

To create stellar-mass-complete samples, we use a modified version of the redshift-dependent r-band apparent magnitude completeness cut of Behroozi et al. (2015):

$$\begin{eqnarray}&&r\lt -0.25-1.9\,\mathrm{log}\left[\displaystyle \frac{{M}_{* }}{{M}_{\odot }}\right]+5\,\mathrm{log}\left[\displaystyle \frac{{D}_{{\rm{L}}}(z)}{10\,\mathrm{pc}}\right],\end{eqnarray} \tag{ 1 }$$

where D_L(z) is the luminosity distance. Behroozi et al. (2015) find that at least 96% of SDSS galaxies within $9.5\lt \mathrm{log}({M}_{* }/{M}_{\odot })\,\lt 10.0$ satisfy ${M}_{r}\lt -0.25-1.9\mathrm{log}({M}_{* }/{M}_{\odot })$, where M_r is the galaxy's Petrosian r-band absolute magnitude. This completeness limit becomes Equation (1) when expressed in terms of redshift and apparent r-band magnitude. Inverting Equation (1) with r = 17.77 gives the minimum stellar mass to which SDSS is ≳96% complete as a function of redshift. Our SDSS samples are complete to a minimum stellar mass of M_* > 10^9.75 M_⊙ at z < 0.0435, which we use as the maximum redshift of our SDSS galaxy samples.

We note, however, that bluer, star-forming galaxies have greater mass-to-light ratios than redder, quiescent galaxies of the same stellar mass. Thus, while 10^9.75 M_⊙ is an appropriate mass-completeness limit for quiescent SDSS galaxies at z < 0.0435, this cut is unnecessarily conservative for star-forming galaxies. Our goal in this study is to probe as wide of a range of galaxy stellar mass as possible, while maintaining an adequately large sample size for robust statistics. Therefore, we determine a lower stellar mass above which star-forming SDSS galaxy samples will be complete.

First, we classify galaxies as star-forming or quiescent based on whether they fall above or below the following redshift-dependent cut in the stellar mass–sSFR plane:

$$\begin{eqnarray}&&\mathrm{log}\left[\displaystyle \frac{\mathrm{sSFR}}{{\mathrm{yr}}^{-1}}\right]=(0.37\,z-0.48)\mathrm{log}\left[\displaystyle \frac{{M}_{* }}{{M}_{\odot }}\right]+3.45\,z-6.10.\end{eqnarray} \tag{ 2 }$$

We obtain this cut by dividing the full SDSS sample into narrow redshift bins containing roughly equal numbers of galaxies and plotting the stellar mass–sSFR distribution in each redshift bin. Next we fit a line of the form $\mathrm{sSFR}=\alpha \,\mathrm{log}({M}_{* }/{M}_{\odot })+\beta$ to the star-forming main sequence (SFMS) in each redshift bin, and then shift this line downward in sSFR to intersect the minimum of the bimodal galaxy stellar mass–sSFR distribution in that bin. We then obtain linear fits to the slope α and intercept β versus the median redshift in each bin to estimate the redshift dependence of each and substitute in these linear expressions for α(z) and β(z) to obtain Equation (2).

To determine an appropriate stellar-mass-completeness cut for star-forming galaxies, we measure in narrow redshift bins the fraction of quiescent galaxies above the Behroozi et al. (2015) stellar mass limit at the median redshift of each bin. We then identify the stellar mass ${M}_{\min }^{\mathrm{SF}}(z)$ at which the same fraction of star-forming galaxies in each bin satisfies ${M}_{* }\geqslant {M}_{\min }^{\mathrm{SF}}(z)$. We find that star-forming galaxies have the same completeness fraction as their quiescent counterparts at stellar masses 0.5–0.6 dex less than the quiescent galaxy limit of 10^9.75 M_⊙ over the redshift range of our sample, with a mean value of 0.55 dex. We therefore adopt a stellar-mass-completeness limit for star-forming galaxies of 10^9.25 M_⊙ at z = 0.0435.

Our resulting stellar-mass-complete SDSS sample contains 41,486 star-forming galaxies with M_* ≥ 10^9.25 M_⊙ and 17,960 quiescent galaxies with M_* ≥ 10^9.75 M_⊙ in the redshift range 0.02 < z < 0.0435.

A primary goal of this paper is to compare the sSFR dependence of galaxy clustering in data and mock catalogs at 0 < z < 1.2. C17 has measured this dependence at 0.2 < z < 1.2 using data from both the PRIsm-MUlti-object Survey (PRIMUS; Coil et al. 2011; Cool et al. 2013) and DEEP2 (Newman et al. 2013) spectroscopic galaxy redshift surveys.

We refer the reader to C17 for full details of the PRIMUS and DEEP2 galaxy samples used in that work—which are the basis for the SDSS and mock galaxy samples used here (see Section 2.4 below)—as well as for additional details about each survey. Briefly, PRIMUS is a low-resolution (R ∼ 40) spectroscopic redshift survey covering ∼9 deg² in seven fields. Conducted with the IMACS instrument (Bigelow & Dressler 2003) on the Magellan I Baade 6.5 m telescope, PRIMUS is the largest spectroscopic faint galaxy redshift survey completed to date. The survey utilized targeting weights to achieve a statistically complete sample of ∼120,000 robust spectroscopic redshifts.

The DEEP2 survey was conducted with the DEIMOS spectrograph (Faber et al. 2003) on the 10 m Keck II telescope and contains ∼17,000 high-confidence redshifts (<95% or Q ≥ 3;  see Newman et al. 2013).

The galaxy samples used in C17 consist of robust (z_quality ≥ 3;  see Coil et al. 2011) PRIMUS redshifts from the CDFS-SWIRE, COSMOS, ES1, and XMM-LSS fields, augmented with Q ≥ 3 DEEP2 redshifts at 0.2 < z < 1.2 from the EGS field. For the remainder of this paper, "PRIMUS data" refers to this combined data set using the PRIMUS and DEEP2 surveys.

C17 estimate stellar masses and SFRs for PRIMUS galaxies by using iSEDfit(Moustakas et al. 2013) to fit spectral energy distributions (SEDs) with the population synthesis models of Bruzual & Charlot (2003), assuming a Chabrier (2003) initial mass function from 0.1 to 100 M_⊙. We refer the reader to Section 2.3 of C17 and to Moustakas et al. (2013) for additional details about PRIMUS stellar mass and SFR estimates.

As the basis for mock galaxy catalogs to compare to observational data, we use snapshots of dark matter N-body simulations at the median redshift of the corresponding data sample.

For z = 0, we use the Bolshoi simulation ⁴ (Klypin et al. 2011), which contains 2048³ particles in a ${(250{h}^{-1}\mathrm{Mpc})}^{3}$ cubic volume, has a dark matter particle mass resolution of 1.35 × 10⁸ M_⊙ h⁻¹, and uses Ω_m = 0.27, Ω_Λ = 0.73, and σ₈ = 0.82. For z = 0.45 and z = 0.9, we use the Bolshoi-Planck simulation (Klypin et al. 2016), which is similar to the Bolshoi simulation but uses Planck cosmological parameters of Ω_m = 0.31 and Ω_Λ = 0.69.

We use the publicly available UniverseMachine (Behroozi et al. 2019) code to populate simulation snapshots at z = 0, 0.45, and 0.9 with synthetic galaxies to create mock galaxy catalogs to which we compare our observational results. In each snapshot, (sub)halos are identified with the publicly available rockstarhalo finder code (Behroozi et al. 2013). UniverseMachine empirically models the dependence of galaxy SFR as a function of halo mass, halo accretion rate, and redshift to predict the star formation histories (SFHs) of individual galaxies over cosmic time and connect those SFHs to the assembly histories of dark matter halos. Halos are populated with synthetic galaxies based on the distributions of a variety of observed galaxy properties across redshifts from z ∼ 0 to z ∼ 10 (summarized in Table 1 of Behroozi et al. 2019).

Table 1. SDSS Galaxy Samples

Run	Sample	Ngal	$\mathrm{log}({M}_{* }/{M}_{\odot })$			$\mathrm{log}($sSFR/yr −1)			Bias a
			min	mean	max	min	mean	max	one-halo	two-halo
SF/Q split	blue	11009	9.75	10.08	10.50	−11.14	−10.23	−8.41	0.78(0.03)	1.19(0.05)
	red	7751	9.75	10.15	10.50	−13.40	−11.69	−10.80	2.26(0.06)	1.97(0.06)
Main-sequence split	dark blue	12582	9.25	9.87	11.35	−11.20	−9.87	−8.41	0.69(0.03)	1.08(0.05)
	light blue	12604	9.25	9.87	11.43	−11.54	−10.37	−9.77	0.92(0.03)	1.33(0.05)
	light red	6712	9.75	10.42	11.59	−12.70	−11.60	−10.80	1.63(0.05)	1.72(0.06)
	red	6732	9.75	10.42	11.75	−13.40	−12.14	−11.40	2.37(0.06)	1.99(0.05)
sSFR cuts	black	10695	9.25	9.63	11.15	−10.00	−9.76	−9.00	0.67(0.03)	1.05(0.05)
	blue	8452	9.25	9.91	11.24	−10.40	−10.19	−10.00	0.82(0.03)	1.22(0.05)
	light blue	4879	9.25	10.09	11.35	−10.80	−10.58	−10.40	1.11(0.04)	1.49(0.05)
	light green	4342	9.75	10.27	11.57	−11.50	−11.16	−10.80	1.51(0.05)	1.68(0.06)
	light red	6134	9.75	10.36	11.55	−12.10	−11.83	−11.50	2.08(0.05)	1.90(0.06)
	red	4462	9.76	10.70	12.00	−13.40	−12.32	−12.10	2.00(0.05)	1.82(0.05)
M*/sSFR grid	purple	7599	9.25	9.47	9.75	−10.00	−9.74	−8.60	0.65(0.03)	1.04(0.05)
	magenta	4401	9.25	9.51	9.75	−10.80	−10.28	−10.00	1.17(0.03)	1.51(0.05)
	black	2846	9.75	9.98	10.40	−10.00	−9.82	−8.41	0.73(0.03)	1.05(0.05)
	blue	6668	9.75	10.06	10.40	−10.80	−10.33	−10.00	0.79(0.03)	1.23(0.05)
	light green	3940	9.75	10.06	10.40	−11.70	−11.29	−10.80	1.87(0.05)	1.89(0.06)
	red	3104	9.75	10.15	10.40	−13.40	−11.97	−11.70	2.82(0.07)	2.11(0.06)
	cyan	2262	10.40	10.61	11.35	−10.80	−10.44	−10.00	0.86(0.03)	1.21(0.05)
	dark green	2037	10.40	10.68	11.43	−11.70	−11.26	−10.80	1.15(0.04)	1.47(0.06)
	light red	5842	10.40	10.75	11.50	−13.36	−12.19	−11.70	1.69(0.04)	1.72(0.05)

Note. All samples span 0.02 < z < 0.0435 and have median redshift z_med ≃ 0.033.

^a One-halo bias is measured on scales of 0.1h⁻¹ Mpc < r_p < 1h⁻¹ Mpc and two-halo bias on scales of 1h⁻¹ Mpc < r_p < 10h⁻¹ Mpc.

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Comparing observations with simulations requires mock galaxy catalogs that have the same joint stellar mass and sSFR distributions, galaxy number density, and line-of-sight positional uncertainty (analogous to redshift error in observational data) as the data set of interest. To achieve this, we use a procedure similar to the one described in Section 3.1 of Berti et al. (2019). Briefly, for each data set, we create a 2D kernel density estimate (KDE) of the joint distribution of galaxy stellar mass versus sSFR. We then select from the relevant synthetic galaxy population a sample with the same 2D distribution as the corresponding observational data. Next we randomly downsample the selected synthetic galaxy population such that it has the same number density as that of the relevant data set at its median redshift, i.e., we match the number density of the z = 0 snapshot to our SDSS sample, and of the z = 0.45 and z = 0.9 snapshots to the number density of PRIMUS data at 0.4 < z < 0.5 and 0.85 < z < 0.95, respectively.

Finally, we add noise to the line-of-sight coordinate r_z of all galaxies in the z = 0.45 and z = 0.9 mocks to simulate the redshift errors of the PRIMUS data set. Specifically, we define a "noisy" line-of-sight coordinate ${r}_{z}^{\mathrm{noisy}}={r}_{z}+{\rm{\Delta }}{r}_{z}$, where Δr_z is drawn from a normal distribution with a dispersion equal to the distance in h⁻¹ Mpc equivalent to the PRIMUS redshift error (σ_z /(1 + z) ≈ 0.0033) at the redshift of the mock. At z = 0.45 ${\sigma }_{{r}_{z}}\simeq 16.4$ h⁻¹ Mpc and at z = 0.9 ${\sigma }_{{r}_{z}}\simeq 21.5$ h⁻¹ Mpc. Redshift errors in SDSS are sufficiently small (York et al. 2000; Blanton et al. 2005) that we do not add additional line-of-sight position noise to the z = 0 mock.

We note that Coil et al. (2011) published estimated PRIMUS redshift errors of σ_z  ≃ 0.005(1 + z), based on a comparison with higher resolution spectroscopic redshifts. However, Behroozi et al. (2019) found that σ_z  ≃ 0.0033(1 + z) is a more accurate estimate of the true PRIMUS redshift errors (see Figure C 6 of Behroozi et al. (2019) and the associated discussion). We test using both σ_z /(1 + z) = 0.0033 and 0.005 to add line-of-sight position uncertainties to the z = 0.45 and z = 0.9 mocks. At both redshifts, the clustering amplitudes and biases of the mocks are in better agreement with PRIMUS data if σ_z /(1 + z) = 0.0033 is used; larger values of σ_z /(1 + z) = 0.005 systematically lower the observed clustering as mock galaxies are overly scattered along the line-of-sight dimension.

Mock galaxies are divided into star-forming and quiescent populations with the following cuts in the stellar mass–sSFR plane:

$$\begin{eqnarray}&&z=0:\mathrm{log}(\mathrm{sSFR})\gt -0.46\mathrm{log}({M}_{* })-6.24;\end{eqnarray} \tag{ 3a }$$

$$\begin{eqnarray}&&z=0.45:\mathrm{log}(\mathrm{sSFR})\gt -0.25\mathrm{log}({M}_{* })-8.06;\end{eqnarray} \tag{ 3b }$$

$$\begin{eqnarray}&&z=0.9:\mathrm{log}(\mathrm{sSFR})\gt -0.19\mathrm{log}({M}_{* })-8.35,\end{eqnarray} \tag{ 3c }$$

where sSFR is in units of yr⁻¹ and M_* is in units of M_⊙. These cuts are determined using an analogous method to that described in Section 2.1 for SDSS galaxies: finding a linear fit to the SFMS in the stellar mass–sSFR plane and shifting this line downward in sSFR so that it intersects the minimum of the bimodal galaxy distribution in this plane.

Two main goals of this paper are (i) to measure the joint dependence of clustering on stellar mass and sSFR in the SDSS, and (ii) compare these measurements in data from both SDSS and PRIMUS to simulations by making analogous measurements in mock galaxy catalogs at 0 < z < 1.2. C17 has measured the clustering dependence of galaxies on stellar mass and sSFR at 0.2 < z < 1.2 with data from the PRIMUS and DEEP2 galaxy redshift surveys ("PRIMUS data" as described above), and their galaxy samples are the basis for both our SDSS samples (see Figure 1 and Table 1) and the mock galaxy samples we create at each redshift (see Figure 2 and 3 and Table 2). We therefore describe C17's samples and the rationale behind them in some detail.

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Table 2. Mock Galaxy Samples

Run	Sample	Ngal	$\mathrm{log}({M}_{* }/{M}_{\odot })$			$\mathrm{log}({sSFR}/{{yr}}^{-1})$			Satellite	Bias a
			min	mean	max	min	mean	max	fraction
z = 0
SF/Q split	blue	49946	9.75	10.08	10.50	−11.07	−10.21	−8.84	0.24	1.21(0.07)
	red	38107	9.75	10.16	10.50	−12.89	−11.68	−10.73	0.46	1.64(0.07)
Main-sequence split	dark blue	55321	9.25	9.88	11.40	−11.16	−9.87	−8.72	0.26	1.25(0.05)
	light blue	55343	9.25	9.89	11.42	−11.41	−10.36	−9.78	0.24	1.19(0.07)
	light red	31422	9.75	10.41	11.65	−12.62	−11.54	−10.73	0.39	1.59(0.04)
	red	31439	9.75	10.41	11.65	−13.24	−12.11	−11.36	0.50	1.76(0.09)
sSFR cuts	black	46367	9.25	9.65	11.15	−10.00	−9.77	−9.0	0.28	1.26(0.04)
	blue	38815	9.25	9.92	11.26	−10.40	−10.19	−10.0	0.23	1.18(0.07)
	light blue	21041	9.25	10.15	11.37	−10.80	−10.57	−10.4	0.21	1.19(0.08)
	light green	19976	9.75	10.28	11.54	−11.50	−11.17	−10.8	0.34	1.44(0.03)
	light red	30157	9.75	10.37	11.46	−12.10	−11.83	−11.5	0.44	1.67(0.05)
	red	17726	9.75	10.67	11.99	−13.24	−12.32	−12.1	0.49	1.82(0.12)
M*/sSFR grid	purple	31878	9.25	9.48	9.75	−10.00	−9.74	−8.72	0.29	1.27(0.04)
	magenta	18243	9.25	9.53	9.75	−10.80	−10.24	−10.0	0.25	1.14(0.07)
	black	13296	9.75	9.98	10.40	−10.00	−9.82	−8.84	0.25	1.22(0.03)
	blue	30953	9.75	10.06	10.40	−10.80	−10.33	−10.0	0.23	1.19(0.08)
	light green	18301	9.75	10.07	10.40	−11.70	−11.30	−10.8	0.41	1.52(0.03)
	red	15265	9.75	10.16	10.40	−12.89	−11.98	−11.7	0.51	1.72(0.11)
	cyan	10660	10.4	10.62	11.37	−10.80	−10.44	−10.0	0.19	1.22(0.09)
	dark green	9714	10.4	10.68	11.47	−11.70	−11.27	−10.8	0.28	1.41(0.03)
	light red	24528	10.4	10.74	11.50	−13.24	−12.16	−11.7	0.45	1.79(0.08)
z = 0.45
SF/Q split	blue	117645	10.5	10.71	11.00	−10.82	−10.01	−8.57	0.21	1.20(0.02)
	red	116897	10.5	10.72	11.00	−12.53	−11.68	−10.74	0.38	1.65(0.05)
Main-sequence split	dark blue	484447	8.5	9.6	10.50	−9.85	−9.19	−8.1	0.28	1.12(0.02)
	light blue	484462	8.5	9.61	10.50	−10.70	−9.66	−8.78	0.34	1.20(0.02)
	light red	116975	10.1	10.64	11.60	−11.84	−11.36	−10.62	0.33	1.54(0.05)
	red	116983	10.1	10.64	11.60	−12.53	−11.88	−11.42	0.44	1.75(0.10)
sSFR cuts	black	119453	8.5	9.17	10.50	−9.00	−8.81	−8.1	0.27	1.04(0.02)
	blue	541553	8.5	9.53	10.50	−9.60	−9.32	−9.0	0.32	1.19(0.01)
	light blue	307452	8.57	9.91	10.50	−10.60	−9.84	−9.6	0.32	1.21(0.02)
	light green	38552	10.0	10.55	11.50	−11.20	−10.98	−10.6	0.31	1.45(0.03)
	light red	139955	10.0	10.56	11.50	−11.80	−11.52	−11.2	0.38	1.58(0.06)
	red	75249	10.0	10.77	11.50	−12.53	−12.00	−11.8	0.44	1.79(0.09)
M*/sSFR grid	purple	194019	8.5	9.1	9.50	−9.20	−8.96	−8.2	0.31	1.10(0.01)
	magenta	226615	8.53	9.26	9.50	−10.20	−9.46	−9.2	0.36	1.19(0.03)
	black	69657	9.5	9.78	10.50	−9.20	−9.04	−8.2	0.24	1.07(0.03)
	blue	463458	9.5	9.95	10.50	−10.20	−9.63	−9.2	0.30	1.18(0.02)
	light green	45038	9.5	10.1	10.50	−11.20	−10.82	−10.2	0.42	1.42(0.05)
	red	95546	9.5	10.21	10.50	−12.20	−11.54	−11.2	0.47	1.57(0.09)
	cyan	90503	10.5	10.72	11.50	−10.20	−9.88	−9.2	0.20	1.17(0.02)
	dark green	57092	10.5	10.84	11.50	−11.20	−10.56	−10.2	0.23	1.35(0.03)
	light red	127046	10.5	10.82	11.50	−12.20	−11.71	−11.2	0.36	1.67(0.07)
z = 0.9
SF/Q split	blue	44044	10.5	10.72	11.00	−10.46	−9.64	−8.13	0.24	1.48(0.03)
	red	62842	10.5	10.76	11.00	−11.90	−11.29	−10.38	0.27	1.82(0.06)
Main-sequence split	dark blue	106422	8.88	9.87	10.50	−9.49	−8.91	−7.98	0.27	1.30(0.04)
	light blue	106430	8.88	9.88	10.50	−10.36	−9.31	−8.25	0.34	1.38(0.05)
	light red	48400	10.1	10.81	11.60	−11.59	−11.07	−10.36	0.23	1.80(0.08)
	red	48410	10.1	10.81	11.60	−11.90	−11.51	−10.82	0.28	1.91(0.08)
sSFR cuts	black	56452	9.0	9.58	11.00	−8.90	−8.66	−8.0	0.24	1.16(0.04)
	blue	158677	9.01	10.04	11.00	−9.60	−9.25	−8.9	0.32	1.40(0.04)
	light blue	39763	9.51	10.55	11.00	−10.20	−9.78	−9.6	0.29	1.45(0.06)
	light green	8828	10.2	10.72	11.69	−10.80	−10.58	−10.2	0.25	1.63(0.07)
	light red	26630	10.2	10.72	11.65	−11.20	−11.03	−10.8	0.24	1.75(0.07)
	red	60417	10.2	10.86	11.69	−11.80	−11.46	−11.2	0.26	1.92(0.09)
M*/sSFR grid	purple	29931	8.88	9.33	9.50	−9.20	−8.67	−8.2	0.26	1.12(0.03)
	black	89738	9.5	9.84	10.50	−9.20	−8.95	−8.2	0.30	1.34(0.06)
	blue	90797	9.5	10.1	10.50	−10.20	−9.44	−9.2	0.34	1.39(0.03)
	light green	9133	9.73	10.33	10.50	−11.20	−10.86	−10.2	0.33	1.58(0.05)
	cyan	48638	10.5	10.79	11.50	−10.20	−9.69	−9.2	0.23	1.50(0.03)
	dark green	27713	10.5	10.81	11.50	−11.20	−10.93	−10.2	0.23	1.76(0.04)
	light red	59321	10.5	10.89	11.50	−11.90	−11.48	−11.2	0.26	1.93(0.10)

Note.

^a These measurements are made on scales of 1h⁻¹ Mpc < r_p < 10h⁻¹ Mpc.

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C17 divide PRIMUS data into two redshift bins, 0.2 < z < 0.7 and 0.7 < z < 1.2. Within each redshift bin, they create subsamples in four different ways to conduct distinct "runs" of their analysis. In their nomenclature, Run 1 ("star-forming/quiescent split") compares the clustering of stellar-mass-complete star-forming and quiescent galaxies within the same stellar mass range.

Run 2 ("main sequence split") divides the star-forming and quiescent populations within a given stellar mass range into two subsamples each: those above and below the star-forming or quiescent main sequence in the stellar mass–sSFR plane. The goal with these samples is to compare the clustering of star-forming (or quiescent) galaxies with above-average sSFRs to those with below-average sSFRs at fixed stellar mass.

C17's Run 3 ("sSFR cuts") again limits the star-forming and quiescent populations in each redshift bin to specified ranges in stellar mass and divides each population into three bins in sSFR. These samples are therefore defined by strict limits in sSFR.

Finally, Run 4 is designed to measure the dependence of clustering on sSFR at fixed stellar mass, as well as the dependence of clustering on stellar mass at fixed sSFR. C17 divide PRIMUS data into nine samples at 0.2 < z < 0.7 and seven samples at 0.7 < z < 1.2 with multiple cuts in both stellar mass and sSFR to create a grid in the stellar mass–sSFR plane. We refer readers to Section 3 in C17 for complete descriptions of the stellar mass, sSFR, and redshift cuts that define their samples.

2.4.1. SDSS Data Samples

2.4.2. Mock Galaxy Samples

To measure projected two-point clustering in SDSS, ω_p(r_p), we use the correl program in UniverseMachine. The program uses the Landy & Szalay (1993) estimator (DD − 2DR + RR) to compute the redshift-space correlation function, ξ(r_p, π), which is then integrated over ∣π∣ < 20 h⁻¹ Mpc to compute the projected correlation function ω_p(r_p). The code uses 10⁶ random points drawn from the same mask region with uniform volume distribution to compute DR, while RR is computed via Monte Carlo integration. Jackknife resampling is used to estimate errors, giving us an estimate for the lower bound of samples with volumes V_eff = 0.3Gpc³. 

In the mock catalogs, to reduce the Poisson errors, we estimate the two-point correlation function ξ(r) of mock galaxy samples by measuring the autocorrelation function (ACF) of all mock galaxies and the cross-correlation function (CCF) between each sample and all galaxies in the mock. We then infer the ACF of each sample as described below.

For ACF and CCF measurements, we use the Halotools (Hearin et al. 2017) function wp_jackknife with the Davis & Peebles (1983) estimator: ξ(r) = DD(r)/DR(r) − 1. For ACF measurements, DD(r) counts pair separations among all galaxies in a given sample, while for CCF measurements, DD(r) is a count of pair separations between the galaxy sample of interest and a "tracer" galaxy sample consisting of the entire mock catalog.

We measure ξ(r) separately both perpendicular to (r_p) and along (π) the mock catalog's line-of-sight dimension, then integrate ξ(r_p, π) over the line of sight to a given value of ${\pi }_{\max }$ to obtain the projected correlation function ω_p(r_p). For each mock, we use the same ${\pi }_{\max }$ value used for ω_p(r_p) measurements in the corresponding data set: ${\pi }_{\max }=20\ {h}^{-1}\,\mathrm{Mpc}$ for the z = 0 mock and SDSS data, ⁵ and ${\pi }_{\max }=40\ {h}^{-1}\,\mathrm{Mpc}$ for the z = 0.45 and z = 0.9 mocks. The latter is consistent with C17 and their clustering measurements of PRIMUS galaxy samples.

The inferred projected ACF ω_p(r_p) for each mock galaxy sample is

$$\begin{eqnarray}&&{\omega }_{{\rm{p}}}({r}_{{\rm{p}}})=\displaystyle \frac{{\omega }_{\mathrm{GT}}^{2}({r}_{{\rm{p}}})}{{\omega }_{\mathrm{TT}}({r}_{{\rm{p}}})},\end{eqnarray} \tag{ 4 }$$

where ω_GT(r_p) is the projected galaxy–tracer CCF, and ω_TT(r_p) is the projected tracer–tracer ACF. Inferring the ACF in this way reduces the error on ω_p(r_p), especially for smaller galaxy samples.

The wp_jackknife function estimates the error of ω_p(r_p) by dividing the mock volume N times along each dimension to define N_j  = (N + 1)³ equal subvolumes and creates the same number of jackknife samples, where each jackknife sample is the entire mock volume excluding one subvolume. The error of ω_p(r_p) is then ${[{\sigma }_{{\omega }_{{\rm{p}}}}^{2}({N}_{j}-1)/{N}_{j}]}^{1/2}$, where ${\sigma }_{{\omega }_{{\rm{p}}}}^{2}$ is the variance of ω_p(r_p) across the jackknife samples.

We measure the absolute bias of both SDSS and mock galaxy samples using the projected correlation function ω_p(r_p) of each sample. Absolute bias is a measure of the clustering strength of a particular galaxy sample compared to that of dark matter and is defined as $\sqrt{{\omega }_{{\rm{G}}}/{\omega }_{\mathrm{DM}}}$, where ω_G and ω_DM are the galaxy and dark-matter-projected correlation functions, respectively, averaged over "two-halo" scales of 1h⁻¹ Mpc < r_p < 10h⁻¹ Mpc. To estimate ω_DM, we use the publicly available code of Smith et al. (2003) to calculate ξ_DM at the median redshift of the relevant galaxy sample, then integrate ξ_DM to the same value of ${\pi }_{\max }$ used for the corresponding galaxy sample.

The relative bias of two galaxy samples is the square root of the ratio of their respective projected correlation functions, averaged over a given length scale, and compares the clustering strengths of the two samples on that scale. We measure the relative bias as a function of scale between pairs of SDSS galaxy samples and pairs of mock galaxy samples, b_rel(r_p), as

$$\begin{eqnarray}&&{b}_{\mathrm{rel}}({r}_{{\rm{p}}})=\sqrt{\displaystyle \frac{{\omega }_{1}({r}_{{\rm{p}}})}{{\omega }_{2}({r}_{{\rm{p}}})}},\end{eqnarray} \tag{ 5 }$$

where ω₁(r_p) and ω₂(r_p) are the projected correlation functions of the two samples. We then take the average of Equation (5) over 0.1h⁻¹ Mpc < r_p < 1h⁻¹ Mpc at a given redshift to estimate the "one-halo" relative bias and average over 1h⁻¹ Mpc < r_p < 10h⁻¹ Mpc to estimate the "two-halo" term of the relative bias.

To estimate the error on absolute and relative bias measurements of SDSS galaxy samples, we calculate the bias b for each jackknife sample and compute the variance of the relevant bias itself across all the jackknife samples. The error of that bias measurement is then $\sqrt{({N}_{j}-1/{N}_{j}){\sigma }_{b}^{2}}$, where N_j is again the total number of jackknife samples and ${\sigma }_{b}^{2}$ is the variance of b across the samples. This method does not use the error on ω_p(r_p) described in the previous section, and instead estimates the uncertainty on the relative bias directly, accounting for cosmic variance across jackknife samples.

Figure 1 shows the stellar mass and sSFR distributions of all SDSS galaxy samples, as well as the projected correlation function ω_p(r_p) of each sample. Table 1 provides the bias on one-halo (0.1h⁻¹ Mpc < r_p < 1h⁻¹ Mpc) and two-halo (1h⁻¹ Mpc < r_p < 10h⁻¹ Mpc) scales for each sample.

The star-forming/quiescent split samples clearly show that quiescent galaxies are more strongly clustered than star-forming galaxies at fixed stellar mass. This confirms previous studies of the dependence of SDSS clustering on galaxy color (e.g., Heinis et al. 2009; Hearin et al. 2014; Watson et al. 2015), often used as a proxy for SFR.

The "main-sequence split" samples in Figure 1 (second column) show that clustering strength is correlated with sSFR at fixed stellar mass within both the star-forming and quiescent populations. Within the star-forming main sequence, galaxies below the sequence are substantially more strongly clustered than galaxies above the sequence. The relative bias between these samples is 1.33 ± 0.02 on one-halo scales and 1.23 ± 0.03 on two-halo scales. We find similar results within the quiescent population, where galaxies below the quiescent sequence are more clustered than galaxies above the sequence, which have higher sSFR at a given stellar mass. This is the first time this has been shown in SDSS and reflects the trends seen at higher redshift in Mostek et al. (2013) and C17.

The third set of samples ("sSFR cuts") displays an anticorrelation between sSFR and clustering strength across the full galaxy population, although the trend is more pronounced for star-forming galaxies. In the lower panel of the third column of Figure 1, the amplitude of ω_p(r_p) increases smoothly as sSFR decreases across five of the six samples, from the highest sSFR sample (dark blue) to the second lowest sSFR sample (light red). The lowest sSFR sample (red) is similar to that of the next lowest sSFR sample (light red); within the errors, ω_p(r_p) is the same for these two samples on both one-halo and two-halo scales. The lack of differentiation of ω_p(r_p) for the two lowest sSFR samples could be due in part to the difficulty of estimating robust SFRs for galaxies with very low star formation rates.

The final set of samples ("stellar mass/sSFR grid") is most easily interpreted by considering separately subsets confined to either a given stellar mass or sSFR bin. Following C17, these samples are used primarily to fill out the range of stellar mass and sSFR ratios over which we explore the dependence of clustering on sSFR at fixed stellar mass and on stellar mass at fixed sSFR in the following sections.

In this section, we compare the clustering measurements in SDSS and PRIMUS data to equivalent measurements in mock galaxy catalogs at z = 0, z = 0.45, and z = 0.9. Table 2 lists the details and two-halo bias measurements of all mock galaxy samples.

The UniverseMachine model is observationally constrained in part by measurements of stellar-mass-complete clustering in SDSS data (at z ∼ 0) and PRIMUS data (at z ∼ 0.45). The model utilizes clustering measurements of star-forming and quiescent galaxies separately, but does not directly incorporate measurements of clustering dependence in narrower bins in sSFR or the intrasequence relative bias previously observed in PRIMUS. A natural question is whether and to what extent does UniverseMachine reproduce the variation in clustering strength with sSFR observed within the star-forming and quiescent galaxy populations in both SDSS and PRIMUS data?

Figure 2 is analogous to Figure 1 and shows the sSFR and stellar mass distributions and projected correlation functions of galaxy samples in the z = 0 mock described above in Section 2. There is excellent agreement between SDSS and the z = 0 mock for the star-forming/quiescent split galaxy samples, largely by design. The amplitude of the ω_p(r_p) one-halo term for quiescent mock galaxies is smaller than that for the analogous SDSS galaxy sample. However, it is known that the clustering amplitude of SDSS galaxies is higher at z < 0.05 (e.g., Figure C5 of Behroozi et al. 2019 and the associated discussion).

Clustering in the SDSS data and the z = 0 mock further diverge when we consider the "main-sequence split" samples. The relative bias within the star-forming z = 0 mock galaxy population—i.e., the bias ratio of galaxies with above-average to below-average sSFRs—is less than unity on both one-halo and two-halo scales, while in the data it is greater than unity. We discuss this reversal below in Section 5.

Two differences between the z = 0 mock and SDSS data are present in the third set of samples ("sSFR cuts"), which divide the galaxy population into six bins in sSFR. In the mock, the clustering strength of the three samples spanning the quiescent sequence correlates with sSFR, especially on one-halo scales. The lowest sSFR quiescent sample (red) has the largest ω_p(r_p), followed by the next lowest sSFR sample (light red). The highest sSFR quiescent sample (light green) is the least strongly clustered of the three quiescent mock samples and splits the difference in ω_p(r_p) amplitude between the other quiescent samples and the three star-forming samples. In contrast, the quiescent "sSFR cuts" samples in SDSS data are less differentiated in terms of relative clustering strength than the corresponding mock samples. The projected correlation functions of the lowest and second lowest sSFR samples agree within the errors, on both one-halo and two-halo scales. As previously discussed in Section 4.1, the lack of differentiation in ω_p(r_p) between the two lowest sSFR samples in SDSS may be at least partly be due to the difficulty of robustly inferring SFRs for SDSS galaxies with the lowest rates of star formation.

At higher redshift, we compare our clustering results in mock catalogs to C17's clustering measurements of analogous galaxy samples in PRIMUS data at 0.2 < z < 0.7 (which we compare to the z = 0.45 mock) and at 0.7 < z < 1.2 (which we compare to the z = 0.9 mock). Interestingly, there are different trends in terms of agreement between the higher redshift mocks and PRIMUS data than for the z = 0 mock and SDSS data.

As expected, the star-forming/quiescent split mock samples at both z = 0.45 (Figure 3) and z = 0.9 agree with what C17 finds in PRIMUS data at 0.2 < z < 0.7 and 0.7 < z < 1.2: quiescent galaxies are more strongly clustered than star-forming galaxies at fixed stellar mass. This agreement is unsurprising, as the star-forming/quiescent split sample results at z ∼ 0.45 from C17 serve as constraints for the UniverseMachine model.

The "main-sequence split" mock samples divide the star-forming and quiescent populations each into two samples with identical stellar mass distributions, allowing us to compare the clustering of star-forming galaxies with above-average and below-average sSFRs, independent of stellar mass, and likewise for quiescent galaxies. We find a correlation between clustering amplitude and sSFR within both the star-forming and quiescent sequences at z = 0.45 and z = 0.9, although the magnitude of the effect is stronger at z = 0.45, particularly for the quiescent population. These results again agree with what C17 find in PRIMUS data: on both one-halo and two-halo scales star-forming galaxies with the highest sSFRs are less clustered than those with sSFRs below the SFMS. Similarly, quiescent galaxies with the lowest sSFRs are more clustered than those with above-average sSFRs.

The third set of galaxy samples ("sSFR cuts") also displays qualitative agreement between the mocks and the corresponding PRIMUS data samples at both 0.2 < z < 0.7 and 0.7 < z < 1.2, although direct comparisons of ω_p(r_p) for the 0.7 < z < 1.2 data samples and z = 0.9 are complicated by the higher noise in the data samples. At 0.2 < z < 0.7, C17 find a general decline in the amplitude of ω_p(r_p) with increasing sSFR, although the two lowest sSFR samples have the same one-halo clustering amplitude within the errors, and two of the three star-forming samples have nearly identical clustering strengths on two-halo scales. In the z = 0.45 mock, the corresponding two star-forming samples also have nearly identical clustering amplitudes on both one-halo and two-halo scales. The highest sSFR star-forming mock sample has the lowest clustering amplitude, which C17 also find for PRIMUS data. While in the z = 0.45 mock we see a clear decline in clustering strength with increasing sSFR across the three quiescent samples, this distinction is less prominent in the corresponding quiescent data samples.

At 0.7 < z < 1.2, the projected correlation functions of the PRIMUS "sSFR cuts" samples are noisier, and a more useful comparison to the corresponding mock is made by comparing the absolute and relative biases, as performed in the following subsections.

Figure 4 shows the absolute bias on two-halo scales of the "main-sequence split" and "sSFR cuts" galaxy samples for both data and mocks at z ∼ 0, z ∼ 0.45, and z ∼ 0.9. At z ∼ 0, there is an overall normalization difference between the data and mock that is not present at higher redshift: the bias values in the mock samples are generally lower than in the corresponding data sample. However, it is known that SDSS data exhibit a clustering excess at z < 0.05 (see Figure C5 of Behroozi et al. 2019), which aligns with the redshift range used here, and is consistent with the bias offsets between the data and z = 0 mock galaxy samples shown in Figure 4.

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As an additional check, we also tested using time-averaged (versus instantaneous) SFRs for the z = 0 mock. While this did slightly improve agreement with the data, the overall trends were the same as using those found using instantaneous SFRs.

Up to the overall normalization difference discussed above, the data and z = 0 mock agree well for quiescent galaxy samples (red, light red, and green points in Figure 4). The one exception is the two "sSFR cuts" samples with the lowest sSFRs (red and light red), which are reversed from the general trend: the lowest sSFR sample is less biased than the next lowest sSFR sample. As discussed in Section 4.1 above, this may be due to the difficulty of robustly estimating SFRs for galaxies with the lowest rates of star formation with sufficient accuracy to meaningfully differentiate between the galaxies in these two samples.

For star-forming galaxy samples (light blue, blue, and dark blue points in Figure 4, highlighted by the two gray circles in the lower-left panels), the data and z = 0 mock do not agree. In the data, the bias decreases monotonically with increasing sSFR, while in the mock, the highest sSFR samples are more biased than star-forming galaxies with lower sSFRs, i.e., the opposite trend. This discrepancy is not seen at the higher redshifts.

Agreement between the data and corresponding mock is strongest at z ∼ 0.45, where the data and mock values match within the errors for every galaxy sample. The z = 0.9 mock samples also agree well with the corresponding data samples, particularly for star-forming galaxies. Within the quiescent population, the lowest sSFR data sample is substantially more biased in the data than in the mock, both when quiescent galaxies are split into two samples above and below the quiescent main sequence in the stellar mass–sSFR plane, and into three samples using simple cuts in sSFR. As noted above, at these higher redshifts, the PRIMUS results are noisier, due to smaller sample sizes.

We now compare the relative biases between pairs of galaxy samples in SDSS and PRIMUS data to the corresponding galaxy sample pairs in mocks. The relative clustering strengths of galaxy samples within the same volume can have a smaller uncertainty than the absolute biases because cosmic variance largely cancels out in relative bias measurements.

Berti et al. (2019) refer to the clustering strength dependence on sSFR within the star-forming or quiescent sequence as ISRB. We adopt that term here to refer to the relative biases of our "main-sequence split" galaxy samples in data and mocks.

The ISRB of quiescent galaxies in the z = 0 mock agrees well with SDSS data within the errors: the relative bias between quiescent mock galaxies with above-average versus below-average sSFRs is 1.24 ± 0.03 on one-halo scales and 1.13 ± 0.03 on two-halo scales, which is  ∼ 9% and  ∼ 4% lower than in the SDSS data, respectively. This agreement disappears for the star-forming population, however, where in z = 0 mock star-forming galaxies with below-average sSFRs are less strongly clustered than those with above-average sSFRs, the opposite of what we find in SDSS data. On both one-halo and two-halo scales, the ISRB within the star-forming z = 0 mock galaxy population is less than unity on both one-halo and two-halo scales, while it is greater than unity in the data.

At higher redshift, there is good qualitative agreement between the mocks and PRIMUS data at both z = 0.45 and z = 0.9. The ISRB C17 find in quiescent and star-forming PRIMUS galaxies on both one-halo and two-halo scales, at both 0.2 < z < 0.7 and 0.7 < z < 1.2, is also present in the z = 0.45 and z = 0.9 mocks, although the magnitude of the ISRB present in the mocks differs somewhat from the corresponding PRIMUS data.

In the z = 0.45 mock, the ISRB values are generally  ∼10% lower than in the data, with the exception of the quiescent one-halo term, which is  ∼5% greater in the mock than in the data. The ISRB in the z = 0.9 mock is  ∼20% to 40% lower than in the data with the exception of the one-halo term for star-forming galaxies, which agrees precisely with the PRIMUS data value.

Following the presentation in C17, Figure 5 shows the relative bias between pairs of galaxy samples in SDSS in PRIMUS data (panels (a) and (b), ⁶ respectively), and between corresponding pairs of galaxy samples in the relevant mock (panels (c) and (d)), as functions of the stellar mass ratio and sSFR ratio of each pair of galaxy samples. In other words, the data points in Figure 5 are a set of unique pairs taken from each of the galaxy samples we create in the stellar mass–sSFR plane. For example, the two "star-forming/quiescent split" samples yield one pair: "red/blue." Relative bias is shown versus stellar mass ratio on the left of each of the four panels, and versus sSFR ratio on the right. The top of each panel shows results on one-halo scales, while two-halo scales are shown in the bottom of each panel.

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All possible galaxy sample pairs are not shown in panel (b) of Figure 5 because C17 include only pairs of PRIMUS data samples for which the one-halo relative bias error is less than 25%. In essence, the larger sample sizes provided by SDSS allow us to fill this parameter space more completely at z ∼ 0. We show results for z ∼ 0 and z ∼ 0.45;  the results at z ∼ 0.9 are very similar.

We find similar trends at z ∼ 0 compared to C17's results at higher redshift. On both one-halo and two-halo scales, the correlation between relative bias and sSFR ratio clearly has less scatter than the trend with stellar mass ratio. A quick comparison of the right columns of panels (a) and (b) may seem to suggest that the correlation of relative bias with sSFR is tighter at higher redshift, but this could also be due to the lower signal-to-noise ratio of PRIMUS data at 0.2 < z < 0.7 compared to SDSS data.

Panels (c) and (d) of Figure 5 present the same analysis in the z = 0 and z = 0.45 mocks. The mocks at both redshifts display a similar relatively tight dependence of relative bias on sSFR ratio that we see in the data. There is little to no correlation between relative bias and stellar mass ratio at either redshift, in qualitative agreement with the z ∼ 0 data. For clarity, Figure 5 does not show results for the z = 0.9 data or mock, but the trends agree with our results at lower redshift.

Again following C17, in Figure 6 we present a way to understand the joint dependence of relative bias on both stellar mass ratio and sSFR ratio in both the data and corresponding mocks. Figure 6 shows the relative bias on two-halo scales of pairs of galaxy samples in the data (left column) and corresponding mock (right column) as a joint function of each pair's stellar mass ratio and sSFR ratio, with the magnitude of the relative bias represented by the color bar in the figure. The dotted lines bracket regions of either fixed stellar mass ratio or sSFR ratio where our samples probe several magnitudes in the ratio of the other parameter. The vertical dotted lines highlight sample pairs with stellar mass ratios of 0.6–2 and sSFR ratios from ∼10⁻³ to ∼10². The relative biases of these sample pairs span the full range of relative bias values observed, from ∼1.6 for the smallest sSFR ratios to ∼0.6 for the largest sSFR ratios. For comparison, the horizontal dotted lines highlight sample pairs with sSFR ratios of 0.9–10, across three orders of magnitude in stellar mass ratio, and show little variation in relative bias across the range of stellar mass ratios probed by our samples. These results confirm at z ∼ 0 the trends observed by C17 in PRIMUS data at 0.2 < z < 1.2, namely that the sSFR ratio is more relevant for determining the relative bias of two galaxy samples than stellar mass ratio. This implies that clustering amplitude is more fundamentally linked to sSFR than to stellar mass.

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Having shown agreement between data and mocks when considering various galaxy samples drawn from the full galaxy population, we now divide each mock galaxy sample into central and satellite components to determine the contribution of each to the trends seen in Figures 5 and 6. Figure 9 shows how the satellite galaxy fraction in the mocks changes with both stellar mass and sSFR. Like the presentation in Figure 7, the set of three panels on the right shows the default mocks, while the leftmost panel shows the "modified" z = 0 mock. The satellite fraction distribution is qualitatively different for star-forming galaxies in the default z = 0 mock compared to the default at higher redshift. In the "modified" z = 0, the satellite fraction distribution more closely resembles that of the higher redshift default mocks: within the SFMS, the satellite fraction is lowest (≲10%) for the highest sSFR galaxies and increases fairly smoothly across the main sequence to  ∼40% for the lowest sSFR star-forming galaxies. In the default z = 0 mock, this trend is reversed across the lower-mass end of the SFMS.

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Results showing the dependence of the relative bias as a function of stellar mass ratio and sSFR ratio separated in the mock samples into centrals and satellites are shown in Figure 10; centrals are represented by magenta circles and satellites by green × symbols. The gray shaded region is the result for all mock galaxies shown in panels (c) and (d) of Figure 5. We focus on the two-halo relative bias (bottom row of Figure 10) as this is the length scale at which central galaxy relative bias is well defined, and therefore also the scale at which meaningful comparisons can be made between centrals and satellites. The same trends are seen on smaller scales (top row), but the errors are larger for the central galaxy relative bias.

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In both the z = 0 and z = 0.45 mocks, the correlation between relative bias and sSFR ratio is driven by central galaxies. This can be clearly seen in the lower right sections of both panels (a) and (b) of Figure 10. In the z = 0 mock, the two-halo relative bias of central galaxies increases from ∼0.6 to ∼1.7 with decreasing sSFR ratio across the nearly five orders of magnitude in sSFR ratio probed here. Over the same range in sSFR ratio, the two-halo relative bias of mock satellites spans the narrower range of ∼0.8 to ∼1.3. At z = 0.45, the two-halo relative bias of mock centrals is again anticorrelated with the sSFR ratio across five orders of magnitude, from ∼0.6 at the largest sSFR ratios to ∼1.8 at the smallest sSFR ratios (lower right of panel (b) in Figure 10). There is a shallower anticorrelation with s slightly larger scatter between the relative bias and sSFR ratio for mock satellites over the same range of sSFR ratios. We also note that in stellar mass ratio space, the division between central and satellite galaxies is not as clean as in sSFR ratio space.

Figure 11 illustrates the difference between the trends for mock centrals and satellites differently (similar to the presentation of Figure 6), showing the relative bias of pairs of central galaxy samples (left column) and pairs of satellite galaxy samples (right column) as a joint function of each pair's stellar mass ratio and sSFR ratio. In other words, to make this figure, we remeasure the clustering and bias of each mock galaxy sample, first keeping only the central galaxies in each sample, then keeping only the satellites. We then calculate the relative bias of pairs of central-only samples and likewise for satellite-only samples. ¹⁰

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The top and bottom rows of Figure 11 show the results for z = 0 and z = 0.45 mocks, respectively. At both z = 0 and z = 0.45, the relative biases of central galaxies span nearly the entire range of the color bar, from magenta at the low end (∼0.6) to orange near the high end (∼1.8). In contrast, the relative biases of satellites at both redshifts span a narrower section of the color bar, from dark blue (∼0.8) to light green (∼1.3). We do not include the z = 0.9 mock in Figure 11, but the results follow the same trends seen in mocks at lower redshift.

In summary, we find that sSFR ratio—not stellar mass ratio—is the key factor influencing the clustering amplitude differences between galaxy samples. Moreover, analysis of the UniverseMachine model indicates that this relationship is driven primarily by central galaxies.

Here we use the simulations from which the mock catalogs are drawn to investigate the extent to which the ISRB is due to central galaxies alone, as opposed to an effect of different satellite fractions above and below each sequence. We use the full simulations instead of mock catalogs to eliminate any possible contributions from effects due to the limitations of the observational galaxy survey data. Our aim is to compare the normalized 3D clustering amplitudes of galaxy samples with different sSFRs, such that any variation in amplitude is purely a prediction of the model and not influenced by added observational effects.

Figure 12 shows the normalized two-point correlation functions (here in three spatial dimensions, not projected onto the plane of the sky) of galaxy samples selected from the full outputs of the UniverseMachine model. These samples use stellar mass and sSFR cuts identical to those that define the mock galaxy samples used in previous sections, but these samples are not subject to the observational effects applied to the simulation to create mock galaxy catalogs that mimic observational data (see Section 2.3).

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The left column of Figure 12 shows the distribution of these galaxy samples in the stellar mass–sSFR plane, as in Figures 2 and 3 above. The remaining panels in each row show ξ(r) for each sample divided by ξ(r) of all galaxies of the relevant type (centrals, satellites, or both types together), which we call ξ_norm. We measure ξ(r) instead of ω_p(r_p) for these galaxy samples because they do not have added line-of-sight position uncertainties.

As seen in Figure 12, the normalized clustering amplitude decreases smoothly with increasing sSFR, both within the star-forming and quiescent populations, and across the entire galaxy population. This trend is present both for all galaxies (centrals and satellites) and for centrals only. A deviation from this trend is seen for star-forming satellite galaxies, where the trend is reversed at both z = 0 and z = 0.45: star-forming satellites with above-average sSFR are more clustered than those with below-average sSFR. This is true for star-forming satellites on both one-halo and two-halo scales, although the difference in the one-halo term is stronger.

To quantify the results of Figure 12, we compute the two-halo ISRB of star-forming and of quiescent mock galaxies, b_rel, as the mean of $\sqrt{{\xi }_{1}(r)/{\xi }_{2}(r)}$ over 1h⁻¹ Mpc < r_p < 10h⁻¹ Mpc, where ξ₁(r) and ξ₂(r) are either for the light blue and dark blue "main-sequence split" samples, respectively, or the red and light red "main-sequence split" samples, respectively, as shown in the figure. We measure b_rel separately for all mock galaxies, mock centrals only, and mock satellites only.

Within the quiescent population, the ISRB at z = 0 is 1.65 for all galaxies, 1.29 for centrals, and 1.17 for satellites. At z = 0.45, the values are lower but the overall trend is the same: quiescent ISRB is greatest for all galaxies (1.42), lower for centrals (1.20), and lowest for satellites (1.10) We emphasize that the two "main-sequence split" samples within each population (star-forming and quiescent) have the same stellar mass distribution by design; clustering amplitude differences are due exclusively to different sample sSFRs.

For star-forming mock galaxies at z = 0, the ISRB is 1.29 for all galaxies, 1.19 for centrals, and 0.93 for satellites. The latter value is less than unity, reflecting the greater clustering of star-forming satellites above the SFMS compared to those below. Similarly, at z = 0.45, the ISRB is 1.10 for all star-forming mock galaxies, 1.09 for centrals, and 0.90 for satellites. While not shown in Figure 12, the trends at z = 0.9 again follow those at lower redshift. These measurements clearly show that the UniverseMachine model predicts an anticorrelation between clustering amplitude and sSFR for central galaxies, similar to what is seen for all galaxies. Further, the reversal of the trend for star-forming satellites (b_rel < 1) means that centrals contribute substantially to the ISRB that is seen for all galaxies.