Black-Hole Growth is Mainly Linked to Host-Galaxy Stellar Mass rather than Star Formation Rate

TL;DR

This study shows that black-hole growth in galaxies is more strongly linked to the host galaxy's stellar mass than to its star formation rate, emphasizing the importance of stellar mass in black-hole evolution.

Contribution

It demonstrates that black-hole accretion rate correlates more strongly with stellar mass than star formation rate, highlighting stellar mass as the key factor in black-hole growth.

Findings

Black-hole accretion rate correlates positively with both SFR and stellar mass.

Black-hole growth is primarily driven by stellar mass rather than star formation activity.

Massive galaxies have higher black-hole fueling efficiency and/or occupation fraction.

Abstract

We investigate the dependence of black-hole accretion rate (BHAR) on host-galaxy star formation rate (SFR) and stellar mass ($M_*$) in the CANDELS/GOODS-South field in the redshift range of $0.5\leq z < 2.0$. Our sample consists of $\approx 18000$ galaxies, allowing us to probe galaxies with $0.1 \lesssim \mathrm{SFR} \lesssim 100\ M_\odot\ \mathrm{yr}^{-1}$ and/or $10^8 \lesssim M_* \lesssim 10^{11}\ M_{\odot}$. We use sample-mean BHAR to approximate long-term average BHAR. Our sample-mean BHARs are derived from the $Chandra$ Deep Field-South 7 Ms observations, while the SFRs and $M_*$ have been estimated by the CANDELS team through SED fitting. The average BHAR is correlated positively with both SFR and $M_*$, and the BHAR-SFR and BHAR-$M_*$ relations can both be described acceptably by linear models with a slope of unity. However, BHAR appears to be correlated more strongly with $M_*$ than SFR. This result indicates that $M_*$ is the primary host-galaxy property related to black-hole growth, and the apparent BHAR-SFR relation is largely a secondary effect due to the star-forming main sequence. Among our sources, massive galaxies ($M_* \gtrsim 10^{10} M_{\odot}$) have significantly higher BHAR/SFR ratios than less-massive galaxies, indicating the former have higher black-hole fueling efficiency and/or higher SMBH occupation fraction than the latter. Our results can naturally explain the observed proportionality between $M_{\rm BH}$ and $M_*$ for local giant ellipticals, and suggest their $M_{\rm BH}/M_*$ is higher than that of local star-forming galaxies. Among local star-forming galaxies, massive systems might have higher $M_{\rm BH}/M_*$ compared to dwarfs.

Tables

Table 1. Table 1 Numbers of Sources in Different Samples

Sample	Low-$z$	High-$z$	Total	Fig(s).
(1)	(2)	(3)	(4)	(5)
All	10057	8164	18221	1 and 2
yr-1 (A)	6384	7541	13925	4
(B)	6445	6669	13114	5 and 8
(A) $\wedge$ (B)	5224	6236	11460	6

Table 2. Table 2 p 𝑝 p -values (Significances) of Partial Correlation Analyses

Relation	Pearson	Spearman	Kendall
BHAR-SFR	${10}^{-6}\left(4.8\sigma \right)$	0.01$\left(2.6\sigma \right)$	$0.34\left(0.9\sigma \right)$
BHAR-$M_{\ast }$	${10}^{-105}\left(22\sigma \right)$	${10}^{-12}\left(7.1\sigma \right)$	$0.01\left(2.6\sigma \right)$
Relation	Pearson	Spearman	Kendall
BHAR-SFR	$0.30\left(1.0\sigma \right)$	$0.23\left(1.2\sigma \right)$	$0.68\left(0.4\sigma \right)$
BHAR-$M_{\ast }$	${10}^{-14}\left(7.5\sigma \right)$	${10}^{-4}\left(3.5\sigma \right)$	$0.06\left(1.9\sigma \right)$

Table 3. Table 3 Numbers of Sources in Different Samples of Star-Forming Galaxies

Sample	Low-$z$	High-$z$	Total	Fig(s).
(1)	(2)	(3)	(4)	(5)
All	8149	6604	14753	N/A
yr-1	5635	6185	11820	9
	5223	5800	11023	10 and 11

Equations

$$⟨L_{\mathrm{X}}⟩=\frac{({\sum }_{\mathrm{detect}}{L}_{\mathrm{X}})+\frac{N_{\mathrm{non}}}{N_{\mathrm{stack}}}{L}_{\mathrm{X},\mathrm{stack}}-{\sum }_{\mathrm{all}}{L}_{\mathrm{X},\mathrm{XRB}}}{N_{\mathrm{detect}}+N_{\mathrm{non}}},$$

$$\begin{array}{rl}⟨\mathrm{BHAR}⟩& =\frac{(1-ϵ)k_{\mathrm{bol}}⟨L_{\mathrm{X}}⟩}{ϵc^2}\\ & =\frac{3.53⟨L_{\mathrm{X}}⟩}{10^{45}\mathrm{erg}{\mathrm{s}}^{-1}}{M}_{\text{sun}}{\mathrm{yr}}^{-1}\end{array}$$

$$\log (⟨\mathrm{BHAR}⟩)=(0.93\pm 0.08)\log (\mathrm{SFR})-(3.85\pm 0.07)$$

$$\log (⟨\mathrm{BHAR}⟩)=\log (⟨\mathrm{SFR}⟩)-(3.89\pm 0.07)$$

$$\log (⟨\mathrm{BHAR}⟩)=(1.10\pm 0.08)\log (M_{\ast })-(14.0\pm 0.8)$$

$$\log (⟨\mathrm{BHAR}⟩)=\log (M_{\ast })-(13.0\pm 0.1)$$

$$\begin{array}{rl}\log (⟨\mathrm{BHAR}& ⟩)=(0.26\pm 0.11)\log (\mathrm{SFR})\\ & +(1.24\pm 0.11)\log (M_{\ast })-(15.5\pm 1.1)\end{array}$$

$$\begin{array}{rl}\log (⟨\mathrm{BHAR}& ⟩)=(0.22\pm 0.08)\log (\mathrm{SFR})\\ & +(1.16\pm 0.09)\log (M_{\ast })-(14.6\pm 0.8),\end{array}$$

$$\frac{M_{\mathrm{BH}}(t_0)}{M_{\ast }(t_0)}\approx \frac{M_{\mathrm{BH}}(t_2)+{\int }_{t_2}^{t_0}\mathrm{BHAR}(t)dt}{M_{\ast }(t_2)+{\int }_{t_2}^{t_0}\mathrm{SFR}(t)dt},$$

$$\frac{M_{\mathrm{BH}}(t_0)}{M_{\ast }(t_0)}\approx \frac{{\int }_{t_2}^{t_0}\mathrm{BHAR}(t)dt}{M_{\ast }(t_2)+{\int }_{t_2}^{t_0}\mathrm{SFR}(t)dt}.$$

$$\frac{M_{\mathrm{BH}}(t_0)}{M_{\ast }}\approx \frac{{\int }_{t_2}^{t_0}\mathrm{BHAR}(t)dt}{M_{\ast }}(\mathrm{elliptical}).$$

$$\begin{array}{rl}\frac{M_{\mathrm{BH}}(t_0)}{M_{\ast }}& \approx \frac{(t_2-t_0)\mathrm{BHAR}}{M_{\ast }}(\mathrm{elliptical})\\ & \approx 10\mathrm{Gyr}\times 10^{-13}\mathrm{y}{\mathrm{r}}^{-1}\\ & \approx 10^{-3}.\end{array}$$

$$\begin{array}{rl}\frac{M_{\mathrm{BH}}(t_0)}{M_{\ast }(t_0)}& \approx \frac{{\int }_{t_2}^{t_0}\mathrm{BHAR}(t)dt}{{\int }_{t_2}^{t_0}\mathrm{SFR}(t)dt}(star-forming)\\ & \sim \frac{\mathrm{BHAR}}{\mathrm{SFR}}\\ & \sim 10^{-5}-10^{-3.5},\end{array}$$

Full text

1. Introduction

The origin of the likely coevolution between supermassive black holes (SMBHs) and their host galaxies remains a fundamental question (e.g., Hopkins et al., 2008; Marulli et al., 2008; Fabian, 2012; Kormendy & Ho, 2013). Observations reveal a linear correlation between star formation rate (SFR) and sample-averaged black-hole accretion rate ($\langle\mathrm{BHAR}\rangle$) for star-forming galaxies (e.g., Chen et al., 2013, C13 hereafter). Also, X-ray-selected active galactic nuclei (AGNs) preferentially reside in star-forming rather than quiescent galaxies for samples with matched stellar mass ($M_{*}$; e.g., Rosario et al., 2013), and optically selected luminous quasars tend to be hosted by strongly star-forming systems (e.g., Harris et al., 2016; Netzer et al., 2016). However, the sample-averaged SFR ($\langle\mathrm{SFR}\rangle$) of the host galaxies of X-ray AGNs do not show a significant dependence on the BHAR in the regimes of low and moderate AGN luminosity, while the potential existence of a positive SFR-BHAR relation at high luminosities is still debatable (e.g., Harrison et al., 2012; Rosario et al., 2012; Barger et al., 2015; Stanley et al., 2015).

To reconcile the apparent discrepancy, Hickox et al. (2014, H14 hereafter) proposed a model in which the long-term ($\sim 100$ Myr) average BHAR traces SFR linearly, but AGN variability hides the BHAR-SFR relation for individual X-ray AGNs (also see Rosario et al. 2013); SFRs are stable over timescales $\gtrsim$ 100 Myr, while AGNs are variable over much shorter timescales. This simple scenario reasonably explains observations, including both the linear $\langle\mathrm{BHAR}\rangle$-SFR relation for star-forming galaxies and the generally flat $\langle\mathrm{SFR}\rangle$-BHAR relation for X-ray-selected AGNs.

The H14 model requires strong AGN variability (by a factor of $\gg 10$) on timescales of $\lesssim 10^{7}$ yr to be commonplace. Although variability studies on the longest available timescales ($\lesssim 10$ yr, rest-frame) do not directly reveal the prevalence of such variability (e.g., Shemmer et al., 2014; Yang et al., 2016), its occurrence on timescales of $10^{2}-10^{7}$ yr is plausible from both observational and theoretical points of view (e.g., Martini & Schneider, 2003; Novak et al., 2011). In fact, some observational evidence suggests the typical AGN-phase time scale is $\sim 10^{5}$ yr, as expected from the chaotic-accretion scenario (e.g., King & Nixon, 2015; Schawinski et al., 2015). Due to the potential existence of such strong variability, the BHAR derived from direct X-ray observations of individual AGNs might not be a reliable indicator of long-term average SMBH growth rate. On the other hand, $\langle\mathrm{BHAR}\rangle$, the average BHAR over a sample of galaxies, serves as a proxy for typical long-term average BHAR of the sample (e.g., C13 and H14). Therefore, $\langle\mathrm{BHAR}\rangle$ provides a useful tool to study SMBH-galaxy coevolution.

Another major motivation of the H14 model is that, in the local elliptical galaxies, the mass of SMBHs ($M_{\rm BH}$) is roughly proportional to the bulge $M_{*}$ (equivalent to host-galaxy $M_{*}$ for ellipticals; see Kormendy & Ho 2013 for a review). If the long-term average BHAR is proportional to SFR for all galaxies, then a natural consequence is that $M_{\rm BH}$ correlates with $M_{*}$ linearly, as long as the accreted mass dominates over the mass of SMBH seeds (e.g., Volonteri, 2010). However, hints have been found of spiral and dwarf galaxies hosting undermassive SMBHs relative to the $M_{\rm BH}$-$M_{*}$ relation derived from ellipticals, although large uncertainties exist (e.g., Greene et al., 2010; Miller et al., 2015; Reines & Volonteri, 2015; Trump et al., 2015; Greene et al., 2016). This behavior is not expected from the H14 model which assumes SFR is the only factor determining long-term average BHAR. Also, simulations indicate that the apparent discrepancy between the $\langle\mathrm{BHAR}\rangle$-SFR and the $\langle\mathrm{SFR}\rangle$-BHAR relations can be produced by the effect of binning on the intrinsic bivariate relationship between BHAR and SFR (e.g., Volonteri et al., 2015), regardless of whether the intrinsic shape of this distribution is produced by an intrinsic long-term BHAR-SFR relation as proposed by H14.

Observations show that the fraction of AGNs above a given luminosity threshold rises steeply toward massive galaxies (e.g., Xue et al., 2010; Aird et al., 2012; Bongiorno et al., 2012; Mullaney et al., 2012a). Furthermore, for $M_{*}$-matched samples, the fraction of galaxies hosting AGNs appears to have no dependence on host-galaxy colors (e.g., Silverman et al., 2009; Pierce et al., 2010; Xue et al., 2010). However, apparent galaxy colors might be a poor indicator of SFR, as high-SFR galaxies might appear red due to significant dust obscuration (e.g., Whitaker et al. 2012; Rosario et al. 2013; and references therein). Therefore, it is still not clear whether $M_{*}$ or SFR is the dominant factor correlated with black-hole accretion.

With the advent of deep ultraviolet-to-infrared (UV-to-IR) observations from surveys such as CANDELS (Grogin et al., 2011; Koekemoer et al., 2011), it has become possible to estimate reliably $M_{*}$ and SFR for the majority population of galaxies with acceptable uncertainties ($\lesssim 0.1$ and $0.2$ dex for $M_{*}$ and SFR, respectively; see Sec. 2.2). The uncertainties on $M_{*}$ and SFR are small compared to the parameter ranges probed (both $\approx 3$ orders of magnitude), and thus are acceptable for our analyses. The 7 Ms Chandra Deep Field-South (CDF-S, covering the whole CANDELS/GOODS-South region; see Luo et al. 2017, L17 hereafter) X-ray survey has achieved unprecedented sensitivities, allowing the derivation of accurate $\langle\mathrm{BHAR}\rangle$ values for the galaxies in CANDELS/GOODS-South. In this paper, we evaluate the dependence of $\langle\mathrm{BHAR}\rangle$ on both SFR and $M_{*}$ for galaxies in CANDELS/GOODS-South. Also, we study the efficiency of SMBH growth compared to SFR for galaxies of different $M_{*}$.

The paper is structured as follows. We describe the sample selection and measurements of SFR, $M_{*}$, and BHAR in Sec. 2. In Sec. 3, we present the results of our analyses. We discuss scientific implications of our results in Sec. 4. We summarize our results in Sec. 5.

Throughout this paper, we assume a cosmology with $H_{0}=70$ km s*-1* Mpc*-1*, $\Omega_{M}=0.3$, and $\Omega_{\Lambda}=0.7$, and a Chabrier initial mass function (Chabrier, 2003). Quoted uncertainties are at the $1\sigma$ (68%) confidence level, unless otherwise stated. SFR and BHAR are in units of $M_{\sun}$ yr*-1*, and $M_{*}$ is in units of $M_{\sun}$, unless otherwise stated.

2. Data Analyses

2.1. Sample Selection

We first select all galaxies with $0.5\leq z<2.0$ and $\mathrm{F160W}<28$ in the CANDELS/GOODS-South catalog (Guo et al., 2013; Santini et al., 2015, S15 hereafter).111$\mathrm{F160W}=28$ is approximately the 5$\sigma$ limiting magnitude of the CANDELS/GOODS-South catalog (Guo et al., 2013). We do not include sources beyond $z=2$, because $M_{*}$ and SFR values estimated from SED fitting (Sec. 2.2) suffer from potential biases in that redshift regime (e.g., Wuyts et al. 2011 and S15). The SEDs of broad-line AGNs often have significant accretion-disk emission besides their starlight. Their SFR and $M_{*}$ measurements from SED fitting have potential large uncertainties (e.g., Bongiorno et al., 2012; Sun et al., 2015). Thus, we exclude the 19 broad-line AGNs reported in the literature (e.g., Mignoli et al., 2005; Ravikumar et al., 2007; Silverman et al., 2010), and discuss the effects of their exclusion in Sec. 3.5.1. Applying these criteria, we select 18221 sources (see Tab. 1), of which 1305 have secure spectroscopic redshifts, and the rest have high-quality photometric redshifts based on up to 17 bands from the UV to IR (S15). Compared to spectroscopic redshifts when available, the photometric redshifts have median uncertainty $|z_{\rm phot}-z_{\rm spec}|/(1+z_{\rm spec})\approx 2\%$ with an outlier (uncertainty $>15\%$) fraction of 3%.

2.2. Stellar Mass and Star Formation Rate

We collect the $M_{*}$ and SFR values for our sources from S15, who presented SED-fitting results of several independent teams. We adopt the median values of $M_{*}$ and SFR from the five available teams (i.e., labeled as 2aτ, 6aτ, 11aτ, 13aτ, and 14a in S15). All five teams employed stellar templates from Bruzual & Charlot (2003) and a Chabrier IMF when performing the SED fitting. Teams 2aτ, 6aτ, 11aτ, and 13aτ assumed an exponentially declining star formation history (SFH); 14a assumed a more flexible SFH (see S15). Teams 2aτ, 6aτ, 11aτ, and 14a adopted Calzetti extinction law (Calzetti et al., 2000); 13aτ adopted a combination of Calzetti and SMC extinction laws. The $M_{*}$ and SFR values estimated by the five teams agree well; their typical deviations from the adopted medians are $\lesssim 0.1$ dex and $\lesssim 0.2$ dex, respectively. These values largely represent uncertainties arising from SED fitting, and additional systematic errors (from, e.g., IMF and SFH assumptions) likely exist. However, such systematic uncertainties should not affect our conclusions qualitatively (see Sec. 3). Fig. 1 shows $M_{*}$ and SFR as functions of redshift, and Fig. 2 shows the $M_{*}$-SFR plane for all our sources. From Figs. 1 (top panel) and 2, X-ray detected sources222These are sources presented in the 7 Ms CDF-S main catalog (Sec. 2.3), formally defined with “binomial no-source probability” $P_{\rm B}<0.007$ (see L17).

are preferentially found among massive galaxies (also see, e.g., Xue et al. 2010; Aird et al. 2012). Our sample is roughly complete for galaxies with $M_{*}\gtrsim 10^{8}\ M_{\sun}$ and $\mathrm{SFR}\gtrsim 0.1\ M_{\sun}\ \mathrm{yr^{-1}}$ (i.e., the $M_{*}$ and SFR regimes mostly probed by our analyses; see Sec. 3). A more detailed discussion of completeness is presented in Sec. 3.5.2.

The rest-frame UV-to-near-IR SEDs ($\approx 0.2-4$ $\mu$m, similar to the wavelength range used by S15 to derive $M_{*}$ and SFR) of X-ray-selected AGNs in the CDF-S are usually dominated by stellar light (see Fig. 9 of Luo et al. 2010; also see, e.g., Xue et al. 2010; Brandt & Alexander 2015). In addition, we have excluded broad-line AGNs (Sec. 2.1), because their $M_{*}$ and SFR measurements from SED fitting could be overestimated (e.g., Ciesla et al., 2015). Therefore, the CANDELS $M_{*}$ and SFR should not have significant biases due to AGN activity. Also, we confirm that AGNs do not have biased SED-based SFRs in comparison with SFRs based on far-IR (FIR) photometry (see below).

As demonstrated by S15, the CANDELS $M_{*}$ values generally have high quality. To evaluate the accuracy of CANDELS SED-based SFRs, which ultimately come from dust-corrected UV luminosities, we compare them with those obtained from FIR photometry. We match our sources with the Herschel/PACS catalog of the PACS Evolutionary Probe (PEP; e.g., Lutz et al. 2011; Magnelli et al. 2013) survey using a 2*′′* matching radius.333We use the 24 $\mu$m-prior PEP catalog due to its good positional accuracy. We follow the method of C13 to derive FIR-based SFRs for the matched sources. Briefly, we convert the PACS band flux to total IR luminosity according to the star-forming galaxy spectral template from Kirkpatrick et al. (2012),444Kirkpatrick et al. (2012) present two templates at $z\sim 1$ and $z\sim 2$, respectively. Here, we use the $z\sim 1$ template following C13, although the $z\sim 2$ template leads to almost the same results. and then we scale the IR luminosity to SFR following the relation from Kennicutt (1998) (modified for our Chabrier IMF; see C13). We use the PACS 100 $\mu$m band for sources at $0.5\leq z<1.3$ and the 160 $\mu$m band at $1.3\leq z<2.0$, requiring photometric $\mathrm{S/N}>5$. Using different bands for different redshift ranges is to sample the SED peak ($\approx 50-80$ $\mu$m, rest-frame; e.g., Kirkpatrick et al. 2012) of cold dust emission, a good proxy for star formation. The FIR-based SFR estimation assumes that dust absorbs most UV photons and reemits IR radiation. It is thus robust for galaxies with relatively high SFR ($\gtrsim 1\ M_{\sun}$ yr*-1*) where dust is often abundant (e.g., Calzetti et al., 2010; Kennicutt & Evans, 2012), and this is indeed our case, since low-SFR galaxies within our redshift range ($0.5\leq z<2.0$) are usually not detected by Herschel.

Fig. 3 compares CANDELS SED-based SFRs with FIR-based SFRs for Herschel-detected galaxies that probe the SFR range of $\approx 10^{0.5}-10^{2.5}\ M_{\sun}$ yr*-1*. CANDELS SFRs generally agree well with FIR-based SFRs without significant systematic bias: the median offset between them is 0.01 dex. The median offset depends on SFR (the blue solid curve in Fig. 3). The offset is small ($\lesssim 0.2$ dex) at low and intermediate SFR, but becomes $\approx 0.3$ dex at $\rm SFR\sim 100$ $M_{\sun}$ yr*-1*. The median offsets for the low-$z$ and high-$z$ bins are 0.07 and 0.16 dex, respectively. A similar systematic offset in the high-SFR (also high-$z$) regime is also found by Wuyts et al. (2011), possibly because dust correction cannot fully recover the intrinsic UV luminosity when the obscuration is very strong. Nevertheless, the systematic errors are smaller than our bin width of SFR (i.e., 0.5 and 1 dex; see Sec. 3), and thus should not affect our results significantly.

For most sources (80%), the SFRs derived by the two methods agree within 0.5 dex (the dashed lines in Fig. 3). The outliers (20%) tend to have FIR-based SFRs higher than SED-based SFRs, possibly because the FIR sample is flux-limited and FIR-luminous outliers are more likely to be detected by Herschel(see also, e.g., Azadi et al. 2015). There are 6% extreme outliers with SFR offsets larger than 1 dex. For these sources, the SFR measurements from different SED-fitting teams are often inconsistent; their typical deviations from the adopted medians are $\approx 0.3-1$ dex, significantly larger than those for the whole sample ($\lesssim 0.2$ dex). The likely failure of SED fitting might be caused by inappropriate model assumptions in SED fitting, e.g., SFH and extinction law. False matches between the CANDELS and FIR catalogs might also be responsible for some of the extreme outliers. Nevertheless, none of these extreme outliers has high X-ray luminosities ($>3\times 10^{42}$ erg s*-1*), and their large SFR uncertainties are unlikely to affect our results qualitatively. Notably, for AGN-dominated X-ray sources with L_{\rm{X}}$$>3\times 10^{42} erg s*-1* (see L17; red symbols in Fig. 3), the CANDELS SFRs do not show a significant systematic bias relative to SFRs derived from FIR photometry: the median offset is 0.01 dex, similar as the value for all Herschel-detected galaxies above. Therefore, the SED-based SFRs are reliable for galaxies with $\mathrm{SFR}\gtrsim 10^{0.5}\ M_{\sun}$ yr*-1*. In the lower-SFR regimes, corrections for dust extinction are generally low or moderate, and SED-based SFRs are generally reliable (e.g., Wuyts et al., 2011; Kennicutt & Evans, 2012).

2.3. Black-Hole Accretion Rate

We use X-ray observations to derive $\langle\mathrm{BHAR}\rangle$ for our sources. We match our 18221 galaxies with the 7 Ms main source catalog for the CDF-S (L17) using a 0.5*′′* matching radius, and 395 X-ray sources are matched.555We use the positions of the L17 optical/near-infrared counterparts rather than the X-ray positions, because the former are more accurate. In the CANDELS/GOODS-South field, there are 704 out of the 1008 X-ray sources in the L17 catalog, and most of them ($674/704=96\%$) have CANDELS counterparts. The remaining 30 sources without counterparts might be faint (F160W band) sources not detected by CANDELS, nearby off-nuclear sources (e.g., ultraluminous X-ray sources), or false X-ray detections (L17). 395 of the 674 X-ray sources with CANDELS counterparts are galaxies in our redshift range ($0.5\leq z<2.0$) without broad lines reported in their spectra.

A total of 259 of the 395 X-ray-detected sources have spectroscopic redshifts. The host-galaxy properties ($z$, $M_{*}$, and SFR) of these X-ray detected sources are shown in Figs. 1 and 2.

We fit their unbinned X-ray spectra (observed-frame 0.5–7 keV) with the Cash statistic (Cash, 1979). We perform the fitting with a standard absorbed power-law model (i.e., $wabs\times zwabs\times powerlaw$ in XSPEC; see Arnaud 1996 for a description of XSPEC) to recover their absorption-corrected X-ray luminosities ($L_{\rm{X}}$, rest-frame 2–10 keV; e.g., Yang et al. 2016). The $wabs$ component accounts for Galactic absorption with absorption column density ($N_{\rm{H}}$) set to $8.8\times 10^{19}$ cm*-2* (Stark et al., 1992). The $zwabs$ component models intrinsic absorption (i.e., $wabs$ at redshift $z$). The normalization of $powerlaw$, intrinsic photon index, and intrinsic $N_{\rm{H}}$ are free parameters in the fitting. The allowed ranges of photon index and $N_{\rm{H}}$ are set to $1.4-2.2$ and $10^{19}-10^{24}$ cm*-2*, respectively. We then obtain the $L_{\rm{X}}$ with XSPEC from the best-fit model parameters. The best-fit $L_{\rm{X}}$ as a function of redshift is shown in Fig. 1 (bottom). Thanks to the great sensitivity of the 7 Ms CDF-S survey, sources with low X-ray luminosities ($\lesssim 3\times 10^{42}$ erg s*-1*) can be detected up to $z=2$. X-ray emission at this low level might not be dominated by AGNs, but could also originate from stellar processes.

Our absorbed power-law model is appropriate for moderately obscured or unobscured AGNs; it might result in unreliable $L_{\rm{X}}$ for Compton-thick AGNs (CTK AGNs; i.e., $N_{\rm H}\gtrsim 10^{24}$ cm*-2*). However, it is generally challenging to identify bona-fide CTK AGNs among the X-ray detected sources in deep fields such as CDF-S due to the limited numbers of counts available, and there is no strong evidence suggesting that CTK AGNs are the dominant population (e.g., Alexander et al. 2013; Brightman et al. 2014; Sec. 3.3 of Brandt & Alexander 2015). We have tested a basic CTK model, $wabs\times zwabs\times pexmon$, on our X-ray detected sources (see Magdziarz & Zdziarski 1995 and Nandra et al. 2007 for $pexmon$; see App. B of Yang et al. 2016 for model parameter settings). Only 4% of sources show statistically significant improvements in fitting compared to our adopted $wabs\times zwabs\times powerlaw$ model, where we use the Akaike information criterion to infer fitting improvement (see Yang et al. 2016 for details). Thus, CTK sources are not likely to be the dominant population in our sample. Depending on model assumptions (e.g., obscuration geometry and viewing angle), the CTK models available often have large uncertainties in the derived physical parameters, e.g., $L_{\rm{X}}$ and $N_{\rm{H}}$ (see, e.g., Murphy & Yaqoob, 2009). Therefore, we do not adopt the spectral-fitting results from the CTK models. A discussion on the effects of X-ray non-detected CTK sources is presented in Sec. 3.5.1.

For sources without X-ray detections in each sample in Sec. 3, we perform a stacking analysis to derive their total $L_{\rm{X}}$ following the procedures of Vito et al. (2016). Briefly, we convert the stacked total count rate of the soft band666Observed-frame 0.5–2 keV. The soft band has larger collecting area and lower background, than the hard band (observed-frame 2–7 keV), e.g., the expected count rate in the soft band is $\approx 2$ times larger than that in the hard band for a $\Gamma=1.8$ power-law spectrum. We have tested stacking the count rate in the hard band. However, the resulting S/N is generally much weaker than that from soft-band stacking, and the hard-band stacked count rates are consistent with zero in many cases. to total $L_{\rm{X}}$ ($L_{\rm X,stack}$) of the stacked sources assuming their median redshift777The median and mean redshifts for our samples (Sec. 3) are similar, and they only differ by $\lesssim 0.02$. and a power-law spectrum with an effective photon index of $\Gamma=1.8$. By setting $\Gamma=1.8$, we assume the stacked signals are mainly from X-ray emission of X-ray binaries (XRBs) and/or AGNs with low obscuration of $N_{\rm H}\lesssim 10^{21.5}$ cm*-1* (see Sec. 6.1 of Lehmer et al. 2016 for a detailed discussion). Indeed, our stacked X-ray fluxes are similar to those expected from XRBs (see Sec. 3.1 and 3.2). Even if we adopt a very flat spectral shape of $\Gamma=1$ (i.e., assume all the stacked X-ray signals are entirely caused by moderately obscured AGNs with $N_{\rm H}\sim 10^{22.5}$ cm*-2*), the resulting $L_{\rm X,stack}$ will be only $\sim 0.3$ dex higher, unlikely to have a large impact on our qualitative results (see Sec. 3). We discuss the potential effects of heavily obscured AGNs that might not be included in our analyses in Sec. 3.5.1. Following Vito et al. (2016), we exclude sources that are at off-axis angles greater than $7.8^{\prime}$ or close to X-ray detected sources (see Vito et al. 2016 for the specific criteria) in the stacking analyses. Those excluded sources represent a real population of galaxies, and thus their contribution to $\langle\mathrm{BHAR}\rangle$ should be included in our analyses. We account for the excluded sources ($\lesssim 25\%$ of the X-ray-undetected sources) in each sample by assuming that their mean $L_{\rm{X}}$ is the same as that of the stacked sources, and scale the $L_{\rm X,stack}$ by multiplying $N_{\rm non}/N_{\rm stack}$ to obtain the total $L_{\rm{X}}$ of all X-ray-undetected sources (Eq. 1).

XRBs and other stellar processes in galaxies also contribute to the observed $L_{\rm{X}}$, and thus we need to subtract their contribution ($L_{\rm{X,XRB}}$).888X-ray emission from XRBs usually dominates over that from other stellar processes. We assume that all non-AGN X-rays are XRB contributions. The $L_{\rm{X,XRB}}$ is estimated as L_{\rm{X,XRB}}$$=\alpha M_{*}+\beta\rm{SFR}, where $\alpha$ and $\beta$ are coefficients as functions of redshift. We adopted the redshift-dependent $\alpha$ and $\beta$ values from model 269 of Fragos et al. (2013) [corrected to our Chabrier IMF following the prescriptions from Longhetti & Saracco (2009) and Madau & Dickinson (2014)] that is preferred by the observations of Lehmer et al. (2016).999We have also tested a simpler $L_{\rm{X,XRB}}$ model (Ranalli et al., 2003) in which $\alpha$ is zero and $\beta$ is a constant (i.e., not dependent on redshift). Our results below only change slightly using this model. Our adopted $\alpha$ and $\beta$ are also consistent with the values from Aird et al. (2016): the diffrences are $\approx 0.1$ dex for both $\alpha$ and $\beta$ in $0.5\leq z<2.0$. The mean AGN $L_{\rm{X}}$ for each sample is calculated as

[TABLE]

where $N_{\rm detect}$, $N_{\rm non}$, and $N_{\rm stack}$ are the numbers of X-ray-detected, undetected, and stacked sources in the sample, respectively. We do not exclude the 18 radio-loud AGNs identified by Bonzini et al. (2013) in our analyses, although excluding them would have only minor effects on our results. Jet-linked X-ray emission might contribute to their $L_{\rm{X}}$, but, at least for the two X-ray brightest radio-loud AGNs, detailed studies do not reveal significant jet-linked X-ray emission (e.g., Iwasawa et al., 2015).

We convert $\langle L_{\rm X}\rangle$ to mean BHAR as

[TABLE]

where we assume a constant bolometric correction factor of $k_{\rm bol}=22.4$ (the median value for the local AGN sample with $L_{\rm X}\approx 10^{41}-10^{45}$ erg s*-1* in Vasudevan & Fabian 2007) and a constant mass-energy conversion efficiency of $\epsilon=0.1$ (e.g., Marconi et al., 2004; Davis & Laor, 2011). We obtain the $1\sigma$ confidence interval as the range between the 16th and 84th percentiles of the bootstrapped $\langle\mathrm{BHAR}\rangle$ distribution. To obtain the $\langle\mathrm{BHAR}\rangle$ distribution, we randomly resample the sources 1000 times. In this routine procedure of bootstrapping, each random resampling includes the same number of sources as the original sample but allowing repetition of sources. We then calculate $\langle\mathrm{BHAR}\rangle$ with Eqs. 1 and 2 for each resampling and obtain the $\langle\mathrm{BHAR}\rangle$ distribution.

From Eq. 2, we obtain $\langle\mathrm{BHAR}\rangle$ from $\langle L_{\rm X}\rangle$. This is because X-ray emission is almost a universal tracer of black-hole accretion (e.g., Gibson et al., 2008). X-rays are also relatively less affected by obscuration compared to the UV/optical bands, and suffer minimal starlight dilution (e.g., Brandt & Alexander, 2015). Nevertheless, there might be uncertainties in the conversion factors ($k_{\rm bol}$ and $\epsilon$) between $\langle\mathrm{BHAR}\rangle$ and $\langle L_{\rm X}\rangle$. The $k_{\rm bol}=22.4$ and $\epsilon=0.1$ assumptions have been widely adopted in previous studies related to black-hole accretion (e.g., Mullaney et al. 2012b and C13). We have also tested applying a luminosity-dependent $k_{\rm bol}$ (Hopkins et al., 2007) for the AGN-dominated sources with L_{\rm{X}}$$>3\times 10^{42} erg s*-1* (e.g., L17), and our results do not change qualitatively. Applying the luminosity-dependent $k_{\rm bol}$ to low-luminosity sources requires careful subtraction of $L_{\rm{X,XRB}}$ for each individual source, but this correction is beyond the scope of our analyses. For simplicity and consistency over all sources, we adopt a constant $k_{\rm bol}$ in our analyses.

Since we include all X-ray detected and non-detected sources (Eq. 1), we are measuring $\langle\mathrm{BHAR}\rangle$ averaged over all galaxies, including systems with both high and low levels of nuclear activity. This is designed to approximate the long-term average BHAR for the entire galaxy sample rather than the instantaneous BHAR for individual AGNs (see Sec. 1).

3. Results

3.1. BHAR Dependence on SFR

We bin our sources in six different SFR intervals (bin width $=0.5$ dex, see Fig. 1) for two different redshift ranges ($0.5\leq z<1.3$ and $1.3\leq z<2.0$), and calculate the $\langle\mathrm{BHAR}\rangle$ for each of the 12 bins which together include 13925 sources (Tab. 1) using Eqs. 1 and 2. In the following analyses, we discard all bins that have fewer than 100 sources to avoid large statistical fluctuations, unless otherwise stated.101010Without this constraint, we can only extend the SFR and $M_{*}$ ranges by $\approx 0.5$ dex (i.e., one bin; see Figs. 4 and 5). Such extended samples have very large uncertainties on $\langle\mathrm{BHAR}\rangle$ ($\sim 1$ dex or only upper limit available) likely caused by statistical fluctuations, since each sample has only $\lesssim 20$ sources. This selection is the main reason why we cannot probe the high-SFR ($\mathrm{SFR}>100$ $M_{\sun}$ yr*-1*) and high-$M_{*}$ ($M_{*}>10^{11}\ M_{\sun}$) regimes (Sec. 3.2).

Fig. 4 displays the results. In general, galaxies with higher SFR have higher $\langle\mathrm{BHAR}\rangle$. We perform a least-$\chi^{2}$ fitting for the $\langle\mathrm{BHAR}\rangle$-SFR relation with a linear model,111111 We employ the Python code, scipy.optimize.curve_fit, to performing the fitting. We use the median SFR of each bin when we perform the fitting. We adopt the mean values of $1\sigma$ upper and lower uncertainties on $\langle\mathrm{BHAR}\rangle$ in the fitting. We do not apply an error to the median SFR for simplicity, since the SFR distribution in each bin usually has a strong non-Gaussian shape. The code estimates the uncertainties on best-fit parameters from the covariance matrix. Median SFR and mean SFR are very close, since sources in each bin have similar SFR. Using the median or mean does not affect our results significantly.

and obtain

[TABLE]

with reduced $\chi^{2}=1.48$ which corresponds to a model-rejection $p$-value of 15%. The slope is consistent with unity. To compare with the H14 model which assumes $\langle\mathrm{BHAR}\rangle$ is proportional to SFR ($\langle\mathrm{BHAR}\rangle=\mathrm{SFR}/3000$), we fix the slope to one and refit the data. This fit results in

[TABLE]

(the solid lines in Fig. 4) and a reduced $\chi^{2}$ of 1.40 ($p$-$\mathrm{value}=17\%$). The best-fit intercept ($-3.89$) is $\approx 0.4$ dex lower than that expected from the H14 model (i.e., $-3.48$; shown as the dotted lines in Fig. 4); possible reasons are explained in Secs. 3.3 and 3.5.1. Our intercept is also similar to the value ($\approx-3.6$) derived from Trump et al. (2015), which is based on optically selected AGNs in the local universe ($z<0.1$).121212 Starting from $\lambda_{\rm Edd}/\mathrm{sSFR}\approx 10^{-2.3}$ in their Fig. 18 (where $\lambda_{\rm Edd}$ is Eddington ratio and sSFR is specific SFR), we obtain $\rm BHAR/SFR\sim 10^{-3.6}$ with the assumption of $\epsilon\sim 0.1$ (Sec. 2.3) and $M_{*}/M_{\mathrm{BH}}\sim 500$ (e.g., Häring & Rix, 2004; Kormendy & Ho, 2013). In general, X-ray emission from XRBs (the dashed lines) is lower compared to that from AGNs, and it is less significant at high SFR. The stacked X-ray emission for individually undetected galaxies is consistent with being entirely due to XRBs.

3.2. BHAR Dependence on $M_{*}$

To investigate the relation between $\langle\mathrm{BHAR}\rangle$ and $M_{*}$, we bin our sources in $M_{*}$ and calculate $\langle\mathrm{BHAR}\rangle$ for each bin. The total number of sources is 13114 (Tab. 1).131313The sample size here is different from that in Sec. 3.1, because the sample here is defined as $10^{8}\leq M_{*}<10^{11}M_{\sun}$ while the sample in Sec. 3.1 is defined as $0.1\leq\mathrm{SFR}<100M_{\sun}$ yr*-1*. The results are shown in Fig. 5. In general, $\langle\mathrm{BHAR}\rangle$ is higher in more massive galaxies, and the fraction of X-ray detected sources rises toward higher $M_{*}$, consistent with previous work (see Sec. 1). In the high-$z$ bins, X-ray emission from AGNs is comparable with that expected from XRBs for galaxies with $M_{*}<10^{10}\ M_{\sun}$, but AGN emission becomes dominant for more massive galaxies. The fact that more massive galaxies have higher $\langle\mathrm{BHAR}\rangle$ is also supported by Fig. 1 and Fig. 2, which demonstrates that most X-ray detections occur in galaxies with $M_{*}\gtrsim 10^{10}\ M_{\sun}$ despite the fact that those massive galaxies are only $\approx 10\%$ of the whole population. Similar to the behavior in Fig. 4 left, X-ray emission from stacked sources is generally comparable to that expected from XRBs. There are several bins ($M_{*}\lesssim 10^{9}\ M_{\sun}$) with only a few X-ray-detected sources ($\lesssim 10$; see Fig. 5 right). In those bins, the $\langle L_{\rm X}\rangle$ contribution from X-ray-detected sources does not dominate over that from stacked sources; thus, the small numbers of detected sources do not cause large statistical fluctuations.

As in Sec. 3.1, we perform a linear fitting to the $\langle\mathrm{BHAR}\rangle$-$M_{*}$ relation. If the slope is allowed to vary, we obtain

[TABLE]

with reduced $\chi^{2}=0.88$ ($p$-$\mathrm{value}=54\%$); if the slope is fixed to unity, we obtain

[TABLE]

with reduced $\chi^{2}=0.94$ ($p$-$\mathrm{value}=50\%$). Therefore, as for the $\langle\mathrm{BHAR}\rangle$-SFR relation, the $\langle\mathrm{BHAR}\rangle$-$M_{*}$ relation can also be described acceptably by a linear relation with a slope of unity. From Fig. 5 left, the $\langle\mathrm{BHAR}\rangle$-$M_{*}$ relation is similar in both redshift ranges. The weak redshift dependence is consistent with the behaviors of the AGN X-ray luminosity function (XLF) and stellar mass function (SMF). Both XLF and SMF of our studied regimes (i.e., $L_{\rm X}\lesssim 10^{44}$ erg s*-1* and $10^{8}\lesssim M_{*}\lesssim 10^{11}M_{\sun}$, respectively) drop slightly by $\approx 0.2$ dex from $z=0.8$ to 1.7, where these redshift values are the medians of the low-$z$ and high-$z$ samples, respectively (see, e.g., Ueda et al. 2014; Tomczak et al. 2014).

In the right panels of Fig. 5, the median SFR values for the bins with $M_{*}\lesssim 10^{10}\ M_{\sun}$ are close to those expected from the star-forming main sequence in the model of Behroozi et al. (2013, B13 hereafter). For massive galaxies with $M_{*}\gtrsim 10^{10}\ M_{\sun}$, our median SFRs are systematically lower than the values expected from the star-forming main sequence. This is likely due to the existence of massive evolved systems in our sample. In fact, after removing the quiescent population, our median SFRs agree much better with the B13 model (see App. A). For both the B13 model and our data, the SFR-$M_{*}$ relation bends at $M_{*}\gtrsim 10^{10}\ M_{\sun}$ (Fig. 5 right), likely due to the depletion of cold gas commonly found in high-$M_{*}$ systems (e.g., Peng et al., 2015).

3.3. BHAR Dependence on Both SFR and $M_{*}$

As shown in Secs. 3.1 and 3.2, $\langle\mathrm{BHAR}\rangle$ has a positive dependence on both SFR and $M_{*}$. However, SFR and $M_{*}$ are not independent properties for star-forming galaxies, which are the major population in our sample. These two properties are positively related to each other via the star-formation main sequence (e.g., Elbaz et al. 2011; also see Fig 2 and the right panels of Figs. 4 and 5). Therefore, it is possible that $\langle\mathrm{BHAR}\rangle$ is fundamentally correlated with one factor, and the observed relation with the other factor is only a secondary effect.

To investigate this possibility, we compare $\langle\mathrm{BHAR}\rangle$ for sources with different $M_{*}$ (SFR) but similar SFR ($M_{*}$). Specifically, we split each sample in Sec. 3.1 into two subsamples, i.e., with $M_{*}\leq\mathrm{med}(M_{*})$ and $M_{*}>\mathrm{med}(M_{*})$, respectively, where $\mathrm{med}(M_{*})$ is the median $M_{*}$ in each sample. For completeness, we also split the samples with highest SFR, although the resulting subsamples have less than 100 sources (but more than 50). The high-$M_{*}$ subsamples have similar typical redshifts compared to the corresponding low-$M_{*}$ subsamples; the differences between their median redshifts are $\approx 0.1$. We then calculate $\langle\mathrm{BHAR}\rangle$ for both subsamples (red downward and blue upward triangles in Fig. 4). In general, the high-$M_{*}$ subsample has significantly higher ($\approx 0.5-1.5$ dex) $\langle\mathrm{BHAR}\rangle$ than its low-$M_{*}$ counterpart. The typical difference between the median $M_{*}$ of the two subsamples is $\approx 0.5$ dex.

Similarly, we also divide each sample in each $M_{*}$ bin (Sec. 3.2) into two subsamples with $\mathrm{SFR}\leq\mathrm{med(SFR)}$ and $\mathrm{SFR}>\mathrm{med(SFR)}$ (Fig. 5). The high-SFR subsamples have slightly higher median redshifts than their low-SFR counterparts; the differences are $\approx 0.25$ and $0.1$ in the low-$z$ and high-$z$ ranges, respectively. This reflects the cosmic evolution of the star-forming main-sequence, i.e., galaxies tend to have higher specific SFR (sSFR, defined as SFR/$M_{*}$) at higher redshift.

As shown in Fig. 5, in the low-$z$ range, the high-SFR subsample generally has higher $\langle\mathrm{BHAR}\rangle$, but for galaxies in the high-$z$ range, the $\langle\mathrm{BHAR}\rangle$ is similar for the two subsamples. Therefore, our results suggest that, at $1.3\leq z<2.0$, $M_{*}$ is likely to be the main physical property correlated with $\langle\mathrm{BHAR}\rangle$, and the observed $\langle\mathrm{BHAR}\rangle$-SFR relation might be a secondary effect caused by the SFR-$M_{*}$ correlation. This is also demonstrated by the comparison between different (sub)samples in Fig. 5 bottom panels. For example, the high-SFR subsample with $10^{9.5}<M_{*}<10^{10}\ M_{\sun}$ has median SFR comparable to those of the two samples with highest $M_{*}$, but its $\langle\mathrm{BHAR}\rangle$ value is much lower than those of the latter. Nevertheless, we cannot rule out possible correlation between $\langle\mathrm{BHAR}\rangle$ and SFR in the high-$z$ range. This is because for bins with $M_{*}<10^{10}\ M_{\sun}$, the $\langle L_{\rm X}\rangle$ from AGNs is low and comparable to $\langle L_{\rm{X,XRB}}\rangle$. Considering the uncertainties associated with, e.g., stacking procedures and XRB modelling, our data are not sufficiently sensitive to differentiate possible $\langle\mathrm{BHAR}\rangle$ differences between high-SFR and low-SFR samples in the low $\langle\mathrm{BHAR}\rangle$ regime (when X-ray emission from AGNs is comparable to or weaker than that from XRBs). It is thus possible that SFR correlates more strongly with $\langle\mathrm{BHAR}\rangle$ at $M_{*}\gtrsim 10^{10.5}\ M_{\sun}$, especially in the low-$z$ range (see the rightmost $M_{*}$ bins in Fig. 5 left); the dependence of $\langle\mathrm{BHAR}\rangle$ on SFR in the high-$M_{*}$ regime is also suggested by some previous studies (e.g., Delvecchio et al., 2015; Rodighiero et al., 2015; Trump et al., 2015). Rosario et al. (2013) suggest that among massive galaxies with $M_{*}\gtrsim 10^{10.5}$ $M_{\sun}$, X-ray AGNs are more prevalent in high-SFR systems (see also, e.g., Azadi et al. 2015). This is consistent with our observations. From the right panels of Fig. 5, in the bins with $10^{10.5}\leq M_{*}<10^{11}$ $M_{\sun}$, the high-SFR subsamples have $\approx 1.5-2$ times higher fractions of X-ray-detected sources than the low-SFR subsamples.

To clarify better whether SFR or $M_{*}$ is more important in the low-$z$ range, we bin our sources with $0.5\leq z<1.3$ over grids of SFR ($0.1\leq\mathrm{SFR}<100\ M_{\sun}\ \mathrm{yr}^{-1}$) and $M_{*}$ ($10^{8}\leq M_{*}<10^{11}\ M_{\sun}$) with the number of sources totaling 5224 (see Tab. 1). We enlarge the bin width to 1 dex to include more sources in each bin and reduce the uncertainties of $\langle\mathrm{BHAR}\rangle$ measurements (see Fig. 2). The larger bin width also makes our results less sensitive to the measurement errors on SFR and $M_{*}$. The results are shown in the upper panel of Fig. 6. As expected, both SFR and $M_{*}$ display positive correlations with $\langle\mathrm{BHAR}\rangle$.

We perform partial-correlation (PCOR) analyses on the results using “pcor” in R.141414The R code pcor is available from http://www.yilab.gatech.edu/pcor.html (e.g., Johnson et al., 2002). In the analyses, we provide pcor with $\log(\langle\rm BHAR\rangle)$, $\log(\rm SFR)$, and $\log(M_{*})$ for sources in each bin (as indicated by the black crosses in Fig. 6). Here, we use a logarithmic scale instead of a linear scale to reduce potential power-law relations to linear relations among the three quantities.

The PCOR analyses are deployed to measure the correlation between $\langle\mathrm{BHAR}\rangle$ and SFR ($M_{*}$) while controlling for the effects of $M_{*}$ (SFR). There are three statistics available in pcor to perform the analyses: one parametric statistic (Pearson) and two non-parametric statistics (Spearman and Kendall). We perform the analyses with all three methods and list the $p$-values from each method in Tab. 2. All $p$-values for the $\langle\mathrm{BHAR}\rangle$-$M_{*}$ relation are significantly smaller than the corresponding $p$-values for the $\langle\mathrm{BHAR}\rangle$-SFR relation, indicating that $\langle\mathrm{BHAR}\rangle$ correlates with $M_{*}$ more strongly than SFR. The parametric method produces $p$-values generally smaller than the non-parametric methods. This is because the parametric method assumes linear relations (see Eqs. 3, 4, 5, and 6; see also Eqs. 7 and 8 below) and uses the input data quantitatively, while the non-parametric methods do not have such assumptions but only use ranks of the input data. Among the non-parametric methods, Spearman’s statistic leads to more significant relations than Kendall’s statistic. The reason is likely that the former uses the value of the difference between two ranks, while the latter is even more conservative, and only considers the sign of the difference (e.g., Feigelson & Babu 2012). A linear regression results in

[TABLE]

with reduced $\chi^{2}=0.51$ (model-rejection $p$-$\mathrm{value}=77\%$; see Footnote 11 for the fitting method).

We also perform the same analyses for sources in both the low-$z$ and high-$z$ ranges together (11460 sources with $0.5\leq z<2.0$ as listed in Tab. 1).151515 We do not analyze the high-$z$ range independently, because this would lead to only three available bins having $>100$ sources (see Sec. 3.1) and well-constrained $L_{\rm{X}}$ (positive $1\sigma$ lower limit); such a small number of bins is not suitable for PCOR analyses.

Fig. 6 (bottom) displays the results. The $p$-values from PCOR analyses are shown in Tab. 2. The $\langle\mathrm{BHAR}\rangle$-$M_{*}$ relation is still significant, in qualitative agreement with the results for the low-$z$ range, but the $\langle\mathrm{BHAR}\rangle$-SFR relation becomes insignificant. To visualize the PCOR analyses, we first fit $\langle\mathrm{BHAR}\rangle$ as a linear function of $M_{*}$ (SFR) in logarithmic space, using the same data as in the PCOR analyses (see Fig. 6 bottom). Then we model the residuals as a linear function of SFR ($M_{*}$), and show the results in Fig. 7. The resulting residual-SFR relation is flat, and its slope is consistent with zero at a $3\sigma$ confidence level. However, the residual-$M_{*}$ relation is steep. Therefore, $\langle\mathrm{BHAR}\rangle$ can largely be described via the relation with $M_{*}$ rather than SFR. Similar analyses have also been performed for the low-redshift bin, and the conclusion is the same.

The linear-fitting ($\langle\mathrm{BHAR}\rangle$ as a function of SFR and $M_{*}$) result is

[TABLE]

similar as in the low-$z$ range. The best-fit reduced $\chi^{2}$ is 2.0 which corresponds to a $p$-$\mathrm{value}=9\%$. The $p$-value is much smaller than the previous value because the errors on $\langle\mathrm{BHAR}\rangle$ from $0.5\leq z<2.0$ ($0.10-0.44$ dex with $\rm median=0.14$ dex) are generally smaller than those from $0.5\leq z<1.3$ ($0.15-0.48$ dex with $\rm median=0.22$ dex) due to the increase of sample size in each bin (Fig. 6).

The above analyses are based on samples including both star-forming and quiescent galaxies, with star-forming galaxies being the major population ($\approx 80\%$). To test if our main conclusion ($\langle\mathrm{BHAR}\rangle$ mainly relates to $M_{*}$) applies for star-forming galaxies alone, we repeat the above analyses with the sample of galaxies near the star-forming main sequence in App. A. Our analyses there show $\langle\mathrm{BHAR}\rangle$ still correlates with $M_{*}$ more strongly than SFR for star-forming galaxies.

3.4. BHAR/SFR Ratio as a Function of $M_{*}$

The ratio BHAR/SFR represents the relative growth between SMBHs and their host galaxies, and thus it has important implications for SMBH-galaxy coevolution. To study its dependence on $M_{*}$, we bin our sources based on $M_{*}$ and derive $\langle\mathrm{BHAR}\rangle$/$\langle\mathrm{SFR}\rangle$ for each bin. The results are displayed in Fig. 8. For both redshift ranges, massive galaxies with $M_{*}\gtrsim 10^{10}\ M_{\sun}$ generally have higher $\langle\mathrm{BHAR}\rangle$/$\langle\mathrm{SFR}\rangle$. This is understandable considering that $\langle\mathrm{BHAR}\rangle$/$M_{*}$ is roughly a constant (Sec. 3.2), while sSFR generally drops for massive galaxies (Fig. 5 right; also see, e.g., Whitaker et al. 2012; Pan et al. 2016).

However, the $\langle\mathrm{BHAR}\rangle$/$\langle\mathrm{SFR}\rangle$ dependence on $M_{*}$ is not observed by Mullaney et al. (2012b, see the blue open points in Fig. 8). This difference is likely because their sample consists of only massive galaxies in a narrow range of $M_{*}$ ($10^{10}\lesssim M_{*}\lesssim 10^{11}\ M_{\sun}$) and their uncertainties are relatively large. A recent study by Rodighiero et al. (2015), based on galaxies with $10^{10}\lesssim M_{*}\lesssim 10^{11.5}\ M_{\sun}$, found that a positive correlation still exists between $\langle\mathrm{BHAR}\rangle$/$\langle\mathrm{SFR}\rangle$ and $M_{*}$, due to their small error bars on $\langle\mathrm{BHAR}\rangle$/$\langle\mathrm{SFR}\rangle$. Our $\langle\mathrm{BHAR}\rangle$/$\langle\mathrm{SFR}\rangle$ values for galaxies having similar $M_{*}$ are slightly lower than those measured in the two studies, likely due to different sample selections and/or the missed accretion power from broad-line AGNs in our analyses (Secs. 2.1 and 3.5.1). Indeed, if we assume the luminous broad-line AGNs have $10^{10}\leq M_{*}<10^{11}\ M_{\sun}$ and include them in our sample, our $\langle\mathrm{BHAR}\rangle$/$\langle\mathrm{SFR}\rangle$ values would be consistent with those in previous studies (see Sec. 3.5.1 for details). The $M_{*}$-dependent ratio of $\langle\mathrm{BHAR}\rangle$/$\langle\mathrm{SFR}\rangle$ provides a possible explanation for the fact that our best-fit intercept of the $\langle\mathrm{BHAR}\rangle$-SFR relation is lower than that expected from the H14 model (see Sec. 3.1). The data used by H14 to estimate the $\langle\mathrm{BHAR}\rangle$-SFR relation are mainly for massive galaxies. Since massive galaxies have higher $\langle\mathrm{BHAR}\rangle$/$\langle\mathrm{SFR}\rangle$ values, the resulting intercept of the $\langle\mathrm{BHAR}\rangle$-SFR relation should be higher. For example, if we only use the high-$M_{*}$ subsamples (blue points in Fig. 4) to derive the $\langle\mathrm{BHAR}\rangle$-SFR relation with unity slope, we would obtain a higher intercept.

The $\langle\mathrm{BHAR}\rangle$/$\langle\mathrm{SFR}\rangle$ ratio in the high-$z$ range is generally lower than that in the low-$z$ range (Fig. 8). This is consistent with global AGN activity and star formation studies. The emissivity of AGNs in our luminosity regime ($L_{\rm X}\lesssim 10^{44}$ erg s*-1*; see Fig. 1) slightly increases from $z\approx 2$ to $z\approx 1$ (e.g., Ueda et al., 2014); meanwhile, the emissivity of star formation drops (e.g., Hopkins 2004 and B13; see also Fig. 5 right). Physically, the redshift evolution might reflect that, at lower redshift, gas in galaxies is more concentrated in the vicinity of SMBHs.

3.5. Reliability Checks

3.5.1 Missed Accretion Power

Luminous X-ray emission is almost a universal tracer of SMBH accretion (e.g., Mushotzky, 2004; Gibson et al., 2008). Since this work is based on the deepest X-ray survey (the 7 Ms CDF-S) and includes individually undetected sources via stacking analyses (Sec. 2.3), we are unlikely to miss a large fraction of black-hole accretion power due to survey sensitivity (e.g., Brandt & Alexander, 2015). Also, our stacked mean X-ray luminosities are similar to the predicted X-ray emission from XRBs (see Secs. 3.1 and 3.2), indicating that most of cosmic accretion power in our redshift range ($0.5\leq z<2.0$) is directly detected in the 7 Ms CDF-S. Due to the small size of CANDELS/GOODS-South (170 arcmin2), our sample will miss AGNs at the bright end of the XLF ($L_{\rm X}\gtrsim 10^{44}$ erg s*-1*; see Fig. 1). For the CANDELS/GOODS-South field ($170\ \rm arcmin^{2}$), the fraction of missed accretion power is $\approx 37\%$ for $0.5\leq z<2.0$, estimated based on the XLF model of Aird et al. (2010). However, those luminous sources are likely to reside in massive systems ($M_{*}\gtrsim 10^{10}\ M_{\sun}$; e.g., Bahcall et al. 1997; Kauffmann et al. 2003; Matsuoka et al. 2014), and thus the bright-end correction is not likely to boost $\langle\mathrm{BHAR}\rangle$ for low-$M_{*}$ galaxies and affect our main conclusions (i.e., massive galaxies have higher $\langle\mathrm{BHAR}\rangle$ and $\langle\mathrm{BHAR}\rangle$/$\langle\mathrm{SFR}\rangle$ than less-massive ones). Moreover, a recent study of luminous quasars also suggests their BHAR might be primarily related to $M_{*}$ rather than SFR (Xu et al., 2015).

A large population of CTK AGNs might exist but show almost no X-ray signal, even in the unprecedentedly deep 7 Ms CDF-S (e.g., see Sec. 3.3 of Brandt & Alexander 2015; Comastri et al. 2015). Their accretion power would be largely missed in our analyses. We have stacked hard-band (observed-frame $2-7$ keV) X-ray images and compared the results with the stacking of the soft band (observed-frame $0.5-2$ keV). We do not find evidence of CTK populations. Nevertheless, it is possible to find signatures of CTK AGNs with more refined analyses (e.g., stacking of very hard narrower bands), and we will perform such analyses in a dedicated paper (B. Luo et al., in preparation). The prevalence of CTK AGNs is predicted by population-synthesis models of the cosmic X-ray background, although these models have significant uncertainties in the regime of very high $N_{\rm{H}}$ (e.g., Gilli et al., 2007; Ueda et al., 2014). The predicted total X-ray emission from CTK AGNs is usually less than that from other AGNs (e.g., Gilli et al. 2007; Treister et al. 2009; also see Buchner et al. 2015 for relevant spectral fitting), consistent with the very-hard X-ray ($\gtrsim 10$ keV) observations of the local universe (e.g., Akylas et al., 2016; Koss et al., 2016). Even if these CTK AGNs predominantly reside in low-$M_{*}$ systems, the total $L_{\rm{X}}$ from these galaxies will be at most the same as that from high-$M_{*}$ galaxies; the average $L_{\rm{X}}$ in low-$M_{*}$ galaxies should be still much lower than for massive galaxies, as the former has much larger number density. Their inclusion is unlikely to make qualitative changes to our main conclusion that $\langle\mathrm{BHAR}\rangle$ is a strong function of $M_{*}$. Nevertheless, the possible existence of the CTK AGN population could increase/decrease the dependence on other galaxy properties at given $M_{*}$. Some observations suggest that CTK AGNs are more prevalent in galaxies with high SFRs and/or that are experiencing major mergers (e.g., Juneau et al., 2013; Kocevski et al., 2015), while the dependence on redshift tends not to be strong (e.g., Buchner et al., 2015).

Nineteen broad-line AGNs have been deliberately excluded from our sample (see Sec. 2.1). The majority of them (14/19) are X-ray luminous with $10^{43}<L_{\rm X}\lesssim 10^{44}$ erg s*-1* (see Fig. 1 bottom). The rarity of broad-line AGNs with $L_{\rm X}<10^{43}$ erg s*-1* is consistent with, e.g., Merloni et al. (2014). The 14 luminous broad-line AGNs make up of 25% of the total AGN population with $L_{\rm X}>10^{43}$ erg s*-1*. This fraction agrees with Fig. 7 of Merloni et al. (2014), which shows the fraction of broad-line AGNs is $\approx 20-40\%$ for $10^{43}\lesssim L_{\rm X}\lesssim 10^{44}$ erg s*-1*. Therefore, we are not missing a significant fraction of broad-line AGNs due to, e.g., low-S/N spectra. Those luminous broad-line AGNs are likely to reside in massive galaxies (e.g., Bahcall et al., 1997; Matsuoka et al., 2014). Indeed, $\approx 90\%$ of our non-broad-line AGNs in the same luminosity regime are hosted by galaxies with $M_{*}>10^{10}\ M_{\sun}$, and these should be physically similar systems as broad-line AGNs following the standard unified AGN model. Hence, including the broad-line AGNs would boost $\langle\mathrm{BHAR}\rangle$ for massive galaxies, and thus would make our main conclusions even stronger. This might also explain why our $\langle\mathrm{BHAR}\rangle$/$\langle\mathrm{SFR}\rangle$ ratio for massive galaxies is lower than the values from previous studies (see Secs. 3.1 and 3.4). Quantitatively, if we include those 14 luminous broad-line AGNs assuming they reside in galaxies with $10^{10}\leq M_{*}<10^{11}\ M_{\sun}$, the $\langle\mathrm{BHAR}\rangle/\langle\mathrm{SFR}\rangle$ values for this $M_{*}$ bin would be $10^{-3.3}$ and $10^{-3.7}$, respectively, in the low-$z$ and high-$z$ ranges. These values are consistent with previous studies (see Fig. 8). Considering the small population and relatively low luminosities ($L_{\rm X}<10^{43}$ erg s*-1*) of the remaining 5 broad-line AGNs, excluding them is not likely to affect our results significantly.

3.5.2 Sample $M_{*}$ and SFR Completeness

In the high-$z$ range, our sample is not complete at $M_{*}\sim 10^{7}\ M_{\sun}$ (Fig. 1 top), and this is the main reason why this study focuses on galaxies with $M_{*}\gtrsim 10^{8}\ M_{\sun}$. To check quantitatively if our sample is complete down to $M_{*}\sim 10^{8}\ M_{\sun}$ in the high-$z$ range, we calculate the comoving number density for the bin with lowest $M_{*}$ and compare it with the model of B13. There are 2550 sources in the bin (see Fig. 5); the comoving volume covered by the CANDELS/GOODS-South field (170 arcmin2; see Guo et al. 2013) is $3.8\times 10^{5}$ Mpc3. Thus, the comoving number density for our galaxies having $10^{8}\leq M_{*}<10^{8.5}\ M_{\sun}$ is $1.3\times 10^{-2}$ Mpc*-3* dex*-1*. This value is roughly consistent with Fig. 3 of B13, indicating our sample is basically complete above $M_{*}\sim 10^{8}\ M_{\sun}$.

A color-dependent completeness issue might exist in our flux-limited sample (e.g., Xue et al., 2010). This is because for a given $M_{*}$, blue galaxies generally have higher optical-to-near-IR luminosities than red galaxies due to their relatively young stellar populations. Therefore, flux-limited optical/IR surveys like CANDELS might miss the red population in the low-$M_{*}$ regimes. However, our sample is not likely to have this issue, since our SFR-$M_{*}$ relation agrees well with the B13 model at the low-$M_{*}$ end (see the right panels of Fig. 5). If our sample were biased to blue (high-SFR) galaxies at the low-$M_{*}$ end, the measured SFR would be significantly above the model value.

The fact that our sample generally does not have color-dependent completeness is not surprising. At the low $M_{*}$ ($\sim 10^{8}\ M_{\sun}$) regime that we probe, the spread in rest-frame colors is relatively small, i.e., most low-$M_{*}$ (thus low-luminosity) galaxies reside in the “blue cloud” rather than the “red sequence” in color-magnitude diagrams (e.g., Schneider, 2014; Martis et al., 2016; Pan et al., 2016). This is broadly consistent with the right panels of Fig. 5, assuming star-forming activity is generally traced by colors. The SFR dispersion is only $\lesssim 0.3$ dex in the bins whose $M_{*}\lesssim 10^{10}\ M_{\sun}$, i.e., most galaxies are located around the star-forming main sequence. In the high-$M_{*}$ regime where the red sequence exists, our sample is also likely to be complete. This is because the comoving number density is $8.5\times 10^{-4}$ Mpc*-3* dex*-1* for the $10^{10.5}\leq M_{*}<10^{11}\ M_{\sun}$ bin in the high-$z$ range; this value is consistent with the stellar-mass function derived in a dedicated study ($9.3^{+1.9}_{-1.7}\times 10^{-4}$ Mpc*-3* dex*-1*; see Tab. 1 of Tomczak et al. 2014), which includes both star-forming and quiescent galaxies.

On the other hand, our lowest-SFR regime ($\mathrm{SFR}\ \sim 0.1\ M_{\sun}$ yr*-1*) corresponds to $M_{*}\sim 10^{8}\ M_{\sun}$ on the star-forming main sequence (see the right panels of Figs. 4 and 5). Since our sample is basically complete above $M_{*}\sim 10^{8}\ M_{\sun}$, it should also be roughly complete above $\mathrm{SFR}\sim 0.1\ M_{\sun}$ yr*-1*. Moreover, our conclusions do not critically depend on the galaxies with $\mathrm{SFR}\sim 0.1\ M_{\sun}$ yr*-1* and $M_{*}\sim 10^{8}\ M_{\sun}$. Thus, even if there are minor completeness issues in our lowest-SFR and/or lowest-$M_{*}$ bins, our results should not be affected materially.

4. Discussion

4.1. SMBH-Galaxy Coevolution

The linear relation between $\langle\mathrm{BHAR}\rangle$ and SFR was previously derived for observations of high-SFR galaxies ($\mathrm{SFR}\gtrsim 10\ M_{\sun}\ \mathrm{yr^{-1}}$, e.g., Symeonidis et al. 2011; C13). For the first time, our results show that this linear relation remains applicable to galaxies with SFRs down to $\sim 0.1\ M_{\sun}\ \mathrm{yr^{-1}}$ in the redshift range of $0.5\leq z<2.0$ (Sec. 3.1). The relation demonstrates that SMBH and galaxy growth broadly track each other over cosmic time. This is consistent with the observational fact that the evolutions of cosmic BHAR and SFR have broadly similar shapes. The normalization of cosmic SFR relative to cosmic BHAR is $\sim 3.7$ dex (see, e.g., Silverman et al. 2008; Aird et al. 2010; Kormendy & Ho 2013), similar to our best-fit intercept of the $\langle\mathrm{BHAR}\rangle$-SFR relation.

The linear $\langle\mathrm{BHAR}\rangle$-SFR relation suggests a simple scenario of coevolution. H14 assumed that, for any individual galaxy, the ratio between the amount of gas accreted by its SMBH and that used to form stars is a universal constant (when averaged over $\sim 100$ Myr). Under this assumption, the $\langle\mathrm{BHAR}\rangle$ for different samples of similar SFR should be similar. However, this assumption appears in contradiction with our observational results. For a given SFR level, the sources with larger $M_{*}$ have significantly higher $\langle\mathrm{BHAR}\rangle$ (Sec. 3.3). In addition, $\langle\mathrm{BHAR}\rangle$ is also related to $M_{*}$ linearly (Sec. 3.2), and $\langle\mathrm{BHAR}\rangle$ is correlated with $M_{*}$ more strongly than SFR as indicated by our PCOR analyses in Sec. 3.3. Therefore, our results suggest that $\langle\mathrm{BHAR}\rangle$ might be intrinsically linked to $M_{*}$, and this $\langle\mathrm{BHAR}\rangle$-$M_{*}$ relation and the star-forming main sequence together might largely cause the observed $\langle\mathrm{BHAR}\rangle$-SFR relation as a secondary effect.

In the analyses of Sec. 3.3, we find when controlling for $M_{*}$ that the dependence of $\langle\mathrm{BHAR}\rangle$ on SFR is relatively weak. We have furthermore checked the $\langle\mathrm{BHAR}\rangle$-sSFR relation for the whole sample, and do not find any significant trend.

4.2. The Physical Link between BHAR and $M_{*}$

It has been well established that X-ray-selected AGNs above a given $L_{\rm{X}}$ threshold are preferentially found in massive galaxies (e.g., Xue et al., 2010; Brandt & Alexander, 2015). This finding is consistent with our results that $\langle\mathrm{BHAR}\rangle$ depends strongly on $M_{*}$, even for SFR-controlled samples (Sec. 3.2 and 3.3). In fact, $M_{*}$, rather than SFR, appears to be the primary factor related to $\langle\mathrm{BHAR}\rangle$.

Massive galaxies with $M_{*}\gtrsim 10^{10}\ M_{\sun}$ have lower sSFR (Sec. 3.4) than less-massive galaxies. If we assume SFR reflects the amount of cold gas available, the decrease of sSFR for massive galaxies indicates the mass fraction of cold gas drops toward high $M_{*}$ (e.g., Saintonge et al., 2016). The cold gas needed for star formation is also likely responsible for fueling black-hole accretion (e.g., Alexander & Hickox, 2012; Vito et al., 2014), while hot-gas accretion could power low-luminosity AGNs that generally have little contribution to total black-hole growth (see, e.g., Fig. 3 of Croton et al. 2006; Heckman & Best 2014; Yuan & Narayan 2014). The $M_{*}$-dependent $\langle\mathrm{BHAR}\rangle$/$\langle\mathrm{SFR}\rangle$ indicates the massive population is generally more efficient in feeding cold gas to their SMBHs (Fig. 8). This could be further broadly interpreted in two possible respects. First, the black-hole fueling efficiency of each galaxy might depend on $M_{*}$ due to several physically plausible causes:

The potential wells in galactic centers are deeper for massive galaxies, making it easier for gas particles to fall into the galaxy center and fuel the SMBH. More specifically, supernova feedback might prevent gas from falling into the galaxy center when the potential well is not sufficiently deep (e.g., Bellovary et al., 2013; Dubois et al., 2015). 2. 2.

Compared to low-$M_{*}$ systems, high-$M_{*}$ ones are more likely to have bars and major mergers (e.g., Melvin et al., 2014; Rodriguez-Gomez et al., 2015) that could induce gas inflow effectively (e.g., Alexander & Hickox, 2012). 3. 3.

Massive galaxies often have more massive SMBHs than low-$M_{*}$ galaxies (e.g., Kormendy & Ho, 2013). Those SMBHs, having a stronger gravitational field, are more capable of accreting gas from their vicinity. In fact, some studies suggest long-term BHAR is proportional to $M_{\rm BH}$, resulting from a universal Eddington-ratio distribution (e.g., Aird et al., 2012; Jones et al., 2016).

Second, the SMBH occupation fraction might drop toward low $M_{*}$ ($M_{*}\lesssim 10^{10}\ M_{\sun}$), and some low-$M_{*}$ galaxies might only host intermediate-mass black holes (IMBHs) with $M_{\rm BH}\lesssim 10^{4}\ M_{\sun}$ (e.g., Volonteri, 2010; Miller et al., 2015; Trump et al., 2015). Due to the Eddington limit, the X-ray emission from accretion onto IMBHs is likely to be much weaker than that from SMBHs. The $\langle\mathrm{BHAR}\rangle$ could thus be diminished for galaxies with lower $M_{*}$.

4.3. Implications for the

$M_{\rm BH}$-$M_{*}$ Relation in the Local Universe

Our results have implications for the $M_{\rm BH}$-$M_{*}$ relation in the local universe, and we illustrate this with some basic arguments below. The $M_{\rm BH}$/$M_{*}$ ratio for a galaxy at $z=0$ can be estimated as

[TABLE]

where $t$ is cosmic time and the subscripts of $t$ indicate redshift. Assuming that the mass of SMBH seeds is small compared to that accreted over cosmic history (e.g., Volonteri, 2010) and most black-hole growth happens at $z\lesssim 2$,161616This is broadly supported by the Sołtan argument, such that $\approx 70\%-80\%$ of total black-hole accretion (including the contribution from luminous quasars) happens at $z\lesssim 2$ (e.g., Ueda et al., 2014; Brandt & Alexander, 2015). High-redshift luminous quasars mostly form SMBHs above $z\sim 2$ (e.g., Wu et al., 2015; Bañados et al., 2016), but they are rare objects and their discussion is beyond the scope of this study. we have $M_{\rm BH}(t_{2})+\int^{t_{0}}_{t_{2}}\mathrm{BHAR}(t)dt\approx\int^{t_{0}}_{t_{2}}\mathrm{BHAR}(t)dt$, and Eq. 9 can be simplified as

[TABLE]

For local giant ellipticals ($M_{*}\gtrsim 10^{11}\ M_{\sun}$), most of their stars are likely to have been formed at $z\gtrsim 2$ (e.g., Chiosi & Carraro, 2002; Siudek et al., 2016). Thus, $M_{*}$ is roughly the same over $z\approx 0-2$, i.e., $M_{*}\equiv M_{*}(t_{0})\approx M_{*}(t_{2})$, and Eq. 10 is approximately171717For galaxy mergers, Eq. 11 is still correct, providing mergers increase $M_{\rm BH}$ and $M_{*}$ proportionally.

[TABLE]

If we assume the BHAR-$M_{*}$ linear relation extends to $M_{*}\gtrsim 10^{11}$ $M_{\sun}$ and has not evolved significantly for $z\lesssim 2$ (Eq. 6 and Fig. 5), Eq. 11 leads to

[TABLE]

Therefore, the $M_{\rm BH}$/$M_{*}$ ratio for giant ellipticals should be approximately a constant (1/1000), similar to the value ($\approx 1/700$) observed by Häring & Rix (2004). Considering that the existence of Compton-thick and broad-line AGNs could boost our observed $\langle\mathrm{BHAR}\rangle$ (see Sec. 3.5.1), the $M_{\rm BH}$/$M_{*}$ ratio might be several times larger and more consistent with the value ($\approx 1/300$) from Kormendy & Ho (2013, but also see ).

The above argument obviously depends on the assumption that giant ellipticals grow their $M_{\rm BH}$ mostly at $z\lesssim 2$. This assumption, although under debate, is supported by observations of submillimeter galaxies (SMGs; see, e.g., Sec. 8.6.7 of Kormendy & Ho 2013). SMGs are likely the high-redshift ($z\sim 2$) progenitors of massive ellipticals (e.g., Casey et al., 2014; Toft et al., 2014), and the growth of their SMBHs tends to lag that of the host galaxies (e.g., Borys et al., 2005; Alexander et al., 2008). Nevertheless, it is also possible to reproduce the local $M_{\rm BH}/M_{*}$ ratio if both the growth of SMBHs and host galaxies take place at $z\gtrsim 2$ and have strong interplay. Simulations show that AGN feedback can keep a tight $M_{\rm BH}$-$M_{*}$ relation when both SMBHs and host galaxies grow at high redshift (e.g., Volonteri et al., 2016).

For local star-forming galaxies, most of their $M_{*}$ is likely to be assembled at $z\lesssim 2$ (e.g., B13). Thus, we have $M_{*}(t_{2})+\int^{t_{0}}_{t_{2}}\mathrm{SFR}(t)dt\approx\int^{t_{0}}_{t_{2}}\mathrm{SFR}(t)dt$, and Eq. 10 becomes

[TABLE]

where we adopt the BHAR/SFR range from Figs. 8 and 11. Our results in Sec. 3.4 show that long-term average BHAR/SFR positively depends on $M_{*}$, and thus $M_{\rm BH}(t_{0})/M_{*}(t_{0})$ is higher for massive galaxies than dwarf galaxies. This prediction is supported by some observations. Miller et al. (2015) and Trump et al. (2015) suggest that dwarf galaxies have a lower black-hole occupation fraction than massive galaxies, implying a generally lower $M_{\rm BH}$/$M_{*}$ for dwarfs. In addition, for some nearby galaxies (e.g., M 33, NGC 205, and NGC 404), studies of nuclear kinematics place tight upper limits on $M_{\rm BH}$, indicating the absence of SMBHs (e.g, Gebhardt et al., 2001; Valluri et al., 2005; Nguyen et al., 2016).

Eqs. 12 and 13 suggest that ellipticals generally have higher $M_{\rm BH}$/$M_{*}$ ratios than star-forming galaxies, consistent with some recent studies (e.g., Reines & Volonteri, 2015; Greene et al., 2016). From the viewpoint of this study, it is understandable why the observed local $M_{\rm BH}$/$M_{*}$ values ($\sim 1/500$; e.g., Häring & Rix 2004; Kormendy & Ho 2013) are much higher than cosmic BHAR/SFR ($\sim 1/5000$; e.g., Silverman et al. 2008; Aird et al. 2010). This is because local $M_{\rm BH}$/$M_{*}$ measurements are mainly based on observations of passive ellipticals (Eq. 12), and cosmic BHAR/SFR is generally linked to $M_{\rm BH}$/$M_{*}$ of star-forming galaxies (Eq. 13).

From Eq. 12, for massive ellipticals, the $M_{\rm BH}$/$M_{*}$ is expected to be lower in the early universe ($z\gtrsim 1$); for star-forming galaxies, it should depend on $M_{*}$ and have relatively weak cosmic evolution according to Eq. 13. As discussed above, observations of SMGs support this scenario. Some studies of high-redshift quasars find higher $M_{\rm BH}$/$M_{*}$ ratios than the local values, not expected in our scheme (e.g., Ho 2007; Merloni et al. 2010; but also see, e.g., Jahnke et al. 2009; Sun et al. 2015). However, the $M_{\rm BH}$/$M_{*}$ measured from quasars might be biased and not representative for the majority of galaxies (e.g., Lauer et al., 2007). Also, large uncertainties often exist in the measurements of both $M_{\rm BH}$ and $M_{*}$ for these quasars (e.g., Bongiorno et al., 2012; Shen, 2013).

5. Summary and Future Prospects

We have studied the dependence of SMBH growth on the SFR and $M_{*}$ of host galaxies at $0.5\leq z<2.0$. Specifically, we compare $\langle\mathrm{BHAR}\rangle$ for samples with different SFR and/or $M_{*}$. Due to the deep multiwavelength data in the CDF-S, we are able to probe black-hole accretion in hosts down to $\mathrm{SFR}\sim 0.1\ M_{\sun}$ yr*-1* and $M_{*}\sim 10^{8}\ M_{\sun}$ with reasonable completeness. Our main results are summarized below:

$\langle\mathrm{BHAR}\rangle$ correlates with SFR linearly (Sec. 3.1). However, for SFR-controlled samples, galaxies with higher $M_{*}$ have higher $\langle\mathrm{BHAR}\rangle$ (Sec. 3.3). Thus, SFR does not appear to be uniquely related to $\langle\mathrm{BHAR}\rangle$. The scenario in which long-term average BHAR is only determined by host-galaxy SFR is over simplified (Sec. 4.1). 2. 2.

$\langle\mathrm{BHAR}\rangle$ is also proportional to $M_{*}$ (Sec. 3.2). In fact, the correlation between $\langle\mathrm{BHAR}\rangle$ and $M_{*}$ is stronger than that between $\langle\mathrm{BHAR}\rangle$ and SFR, suggesting $M_{*}$ as the primary host-galaxy property related to SMBH growth. This result also holds for the star-forming population alone (App. A). The observed $\langle\mathrm{BHAR}\rangle$-SFR correlation might be largely a secondary effect due to the existence of the star-formation main sequence (Sec. 4.1). 3. 3.

Massive galaxies ($M_{*}\gtrsim 10^{10}\ M_{\sun}$) have higher $\langle\mathrm{BHAR}\rangle$/$\langle\mathrm{SFR}\rangle$ ratios than their less-massive counterparts (Sec. 3.4), suggesting that they have higher black-hole fueling efficiency and/or SMBH occupation fraction (Sec. 4.2). 4. 4.

Our results can naturally explain the observed $M_{\rm BH}$-$M_{*}$ relation for local giant ellipticals, and indicate that they have higher $M_{\rm BH}$/$M_{*}$ than their progenitors in the earlier universe (Sec. 4.3). Also, our results predict that $M_{\rm BH}$/$M_{*}$ for giant ellipticals is higher than that for star-forming galaxies in the nearby universe. Among local star-forming galaxies, the $M_{\rm BH}$/$M_{*}$ values for massive galaxies are likely to be higher than those for dwarfs.

In the future, this study could be extended to galaxies with larger $M_{*}$ (SFR) by compiling a large number of luminous galaxies (Sec. 3.1). To perform this investigation, analyses based on multiwavelength surveys of wider fields, e.g., COSMOS, XMM-LSS, and Stripe 82, are needed. In addition, it is possible to extend this study to higher redshift using the CDF-S field, but this approach will require SED fitting that can eliminate potential high-redshift biases for SFR and $M_{*}$ measurements (Sec. 2.2). It would also be worthwhile to derive quantitative black-hole fueling efficiency and/or SMBH occupation-fraction estimates as a function of $M_{*}$, based on the $M_{*}$-dependent $\langle\mathrm{BHAR}\rangle$/$\langle\mathrm{SFR}\rangle$ (Sec. 3.4). Future work could study the BHAR for giant ellipticals by including morphological information (Sec. 4.3) and the connection between BHAR and host-galaxy gas content by using ALMA observations.

Acknowledgements

We thank the referee for helpful feedback that improved this work. We thank Robin Ciardullo, Ryan Hickox, Luis Ho, Paola Santini, and Xue-Bing Wu for helpful discussions. G.Y., C.-T.J.C., F.V., and W.N.B. acknowledge financial support from CXC grant GO4-15130A. D.M.A. acknowledges the Science and Technology Facilities Council (STFC) for support through grant ST/L00075X/1. B.L. acknowledges support from the National Natural Science Foundation of China grant 11673010 and the Ministry of Science and Technology of China grant 2016YFA0400702. M.Y.S. and Y.Q.X. acknowledge support from the National Thousand Young Talents program, the 973 Program (2015CB857004), NSFC-11473026, NSFC-11421303, the CAS Strategic Priority Research Program (XDB09000000), the Fundamental Research Funds for the Central Universities, and the CAS Frontier Science Key Research Program (QYZDJ-SSW-SLH006). F.E.B. acknowledges support from CONICYT-Chile Basal-CATA PFB-06/2007 and FONDECYT Regular 1141218, and the Ministry of Economy, Development, and Tourism’s Millennium Science Initiative through grant IC120009, awarded to The Millennium Institute of Astrophysics, MAS. J.-X.W. acknowledges support from NSFC-11233002 and 973 program (2015CB857005). This project uses Astropy (a Python package; see Astropy Collaboration et al. 2013).

Appendix A Analyses for Star-Forming Galaxies

Here we perform the same analyses as in Sec. 3 but for star-forming galaxies only, in order to check if our conclusions also apply for this population alone. We define star-forming galaxies as sources having sSFR within 5 times (i.e., 0.7 dex) of the median sSFR of the whole sample in the corresponding redshift range.181818Different studies often adopt different empirical definitions of star-forming galaxies. Here we adopt a similar definition as Elbaz et al. (2011), i.e., the galaxies with sSFR around a typical value. We adopt a wider sSFR range than Elbaz et al. (2011) when defining the main sequence, mostly because our SED-based SFR estimations have larger uncertainties than the FIR-based SFR estimations in Elbaz et al. (2011, see our Sec. 2.2).

The median sSFRs for the low-$z$ and high-$z$ samples are $10^{-8.94}$ and $10^{-8.72}$ yr*-1*, respectively. The numbers of star-forming galaxies in the low and high redshift ranges are 8149 and 6604, respectively (see Tab. 3).

Similar to Sec. 3, we bin sources based on SFR ($M_{*}$), and split each sample based on $M_{*}$ (SFR). For straightforward comparison with our results for all galaxies, we allow bins with numbers of sources less than 100 in our analyses. Figs. 9 and 10 show the results. Similar to the results in Sec. 3.1 and 3.2, sources with higher SFR ($M_{*}$) generally have higher $\langle\mathrm{BHAR}\rangle$. However, due to the apparent non-linearity, the linear models in Sec. 3.1 and 3.2 result in unacceptable fitting quality, and thus we do not show the fitting results here. From Fig. 9 (Fig. 10), the non-linearity mainly arises from the apparent steep rise of $\langle\mathrm{BHAR}\rangle$ above the threshold of $\mathrm{SFR}\sim 10^{0.5}M_{\sun}$ yr*-1* ($M_{*}\sim 10^{10}M_{\sun}$ yr*-1*). The steep change might be caused by statistical fluctuations atop a gradual rise, but it could also indicate an intrinsic threshold of SFR ($M_{*}$) above which AGN X-ray emission becomes dominant over that of XRBs for star-forming galaxies. Larger samples are needed to differentiate the two possibilities. The SFR-$M_{*}$ relation for star-forming galaxies in Fig. 10 (right) is more similar to that of B13 compared to that in Fig. 5 at the high-$M_{*}$ end (see Sec. 3.2).

From Fig. 9, the high-$M_{*}$ subsamples generally have higher $\langle\mathrm{BHAR}\rangle$ than the corresponding low-$M_{*}$ subsamples. From Fig. 10, the two subsamples with different SFRs generally have similar $\langle\mathrm{BHAR}\rangle$. These results qualitatively agree with our major conclusion that $\langle\mathrm{BHAR}\rangle$ correlates with $M_{*}$ more strongly than SFR (see Sec. 3.3). In Fig. 10, the low-SFR subsamples of massive galaxies ($M_{*}\gtrsim 10^{10}$ $M_{\sun}$) even appear to have higher $\langle\mathrm{BHAR}\rangle$ than their high-SFR counterparts. However, the difference is not statistically significant, and larger samples are needed to clarify this point. From PCOR analyses (see Sec. 3.3), the parametric method (Pearson) results in $\approx 2-3\sigma$ significances of the $\langle\mathrm{BHAR}\rangle$-$M_{*}$ relation in the two redshift ranges in Tab. 2, while the method shows the $\langle\mathrm{BHAR}\rangle$-SFR relation is insignificant in both redshift ranges. However, this parametric method models correlations linearly, which is likely not appropriate considering the apparent non-linearity in Figs. 9 and 10. The non-parametric methods (Spearman and Kendall) do not assume linearity but have less statistical power (see Sec. 3.3). These methods cannot reveal high significances for either the $\langle\mathrm{BHAR}\rangle$-SFR or $\langle\mathrm{BHAR}\rangle$-$M_{*}$ relations. The reasons are likely to be the reduced sample size (especially the reduced number of AGNs; see Fig. 2) and the reduced coverage of the SFR-$M_{*}$ plane.

Fig. 11 shows the ratio between $\langle\mathrm{BHAR}\rangle$ and $\langle\mathrm{SFR}\rangle$ for the star-forming galaxies. As in Fig. 8 for all galaxies, the ratio is higher for massive galaxies with $M_{*}\gtrsim 10^{10}\ M_{\sun}$. As expected from Fig. 10, there is a sudden “jump” of $\langle\mathrm{BHAR}\rangle$/$\langle\mathrm{SFR}\rangle$ at $M_{*}\sim 10^{10}\ M_{\sun}$, probably indicating a physical $M_{*}$ threshold above which AGN activity starts to become very strong. Compared to the corresponding bins in Fig. 8, $\langle\mathrm{BHAR}\rangle$/$\langle\mathrm{SFR}\rangle$ values are slightly lower in general, although the trend is weak and within the error bars. This is expected, as the quiescent population has lower $\langle\mathrm{SFR}\rangle$ but similar $\langle\mathrm{BHAR}\rangle$ (our main conclusion) compared to the star-forming population at given $M_{*}$.