X-RAY DETECTED ACTIVE GALACTIC NUCLEI IN DWARF GALAXIES AT 0 < z < 1

For reliable stellar masses, we turn to the NMBS (see Whitaker et al. 2011, for a complete description), which is a near-infrared imaging survey of part of the AEGIS field. Since this survey splits the broadband JHK bands into seven medium-band filters, there is improved modeling of the spectral energy distribution, which leads to more dependable stellar masses (van Dokkum et al. 2009). The $27^{\prime} .6\times 27^{\prime} .6$ NMBS survey region is centered at $\alpha ={14}^{{\rm{h}}}{18}^{{\rm{m}}}{00}^{{\rm{s}}}$, $\delta =+52^\circ {36}^{{\rm{m}}}{07}^{{\rm{s}}}$ (J2000; van Dokkum et al. 2009). The K-band 90% limiting magnitude is $22.5\ \mathrm{AB}\ \mathrm{mag}$ and the 50% limiting magnitude is 23.6 AB mag (Whitaker et al. 2011). The NMBS sources were K-band selected using SExtractor (Bertin & Arnouts 1996), and the stellar population parameters, including star-formation rates (SFRs), were derived with FAST (Kriek et al. 2009) using the Bruzual & Charlot (2003) models and an exponentially declining star-formation history. The Bruzual & Charlot (2003) models use the Chabrier (2003) initial mass function. To ensure more accurate stellar masses, the NMBS used spectroscopic redshifts from the DEEP2 survey, which has a limit of $R\lt 24.1$ mag (Davis et al. 2002; Steidel et al. 2003), when available. For our purposes, accurate stellar masses are paramount, hence we restrict ourselves to the sources with spectroscopic redshifts. In addition, we restrict the galaxies to those with ${z}_{\mathrm{spec}}\lt 1$ to match the mass sensitivity of the NMBS. Of the 27652 sources identified by the NMBS, we find 642 within our mass and redshift ranges that have spectroscopic redshift measurements.

To ensure that all of our sources are dwarf galaxies, rather than tidal tails mistakenly identified by the NMBS, we require morphologies of our sources. HST surveyed a $0.197\ {\deg }^{2}$ region of the AEGIS field using the Advanced Camera for Surveys (ACS; Davis et al. 2007). Using the HST data, we discard 37 of the galaxies within our mass and redshift ranges because they are mergers. We exclude these mergers because of the possibility of inaccurate mass measurements. This leaves us with 605 sources selected from the NMBS and HST data.

The 605 galaxies selected are shown, binned by mass and redshift, in Figure 2. The masses range from $\sim 5\times {10}^{7}\,{M}_{\odot }$ to $3\times {10}^{9}\,{M}_{\odot }$. The SFRs vary from $\sim 3\times {10}^{-4}$ to $10\ {M}_{\odot }\ {\mathrm{yr}}^{-1}$, with a median SFR = 0.24 ${M}_{\odot }\ {\mathrm{yr}}^{-1}$.

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We now identify the dwarf galaxies from our sample that show significant X-ray emission by using the available X-ray data within the AEGIS field. The Chandra X-ray Observatory has observed $\sim 0.7\ {\deg }^{2}$ of AEGIS with the Advanced CCD Imaging Spectrometer (ACIS) as part of XDEEP2 (Goulding et al. 2012, see also Nandra et al. 2005; Laird et al. 2009; Nandra et al. 2015). To ensure the most reliable measurements, we apply the latest calibration files using the CIAO software (version 4.7) and CALDB 4.6.5 (Fruscione et al. 2006).

Our X-ray analysis consists of performing forced aperture photometry at the optical positions of our selected galaxies. Some of our selected galaxies have X-ray data from different, overlapping subregions within AEGIS. However, we only use the X-ray data from the region with an aim-point closest to each source. This is because we wish to limit the amount of contamination from diffuse emission, and the Chandra point-spread function (PSF) deteriorates with distance from the aim-point. We define the X-ray source region as the 90% enclosed energy radius around the optical source position using the CIAO psf module. This is done for the soft-band (SB; 0.5–2 keV), the hard-band (HB; 2–7 keV), and the full-band (FB; 0.5–7 keV). The 90% enclosed energy radii range from ∼1'' to 15'' for most of our sources. We define the background region to be a square with sides that are at least four times as large as the radius of the source region. We find that 23 of the 605 dwarf galaxies selected from the NMBS have a second spatially coincident galaxy within the 90% enclosed energy radius. The X-ray photons associated with these neighbors cannot be reliably disentangled from the target dwarf galaxy, and thus these 23 sources are discarded from the sample.

Note that our method here differs from that of traditional X-ray catalogs created independently of multi-wavelength source positions. Using X-ray data alone to detect a source necessitates a high cut in X-ray signal-to-noise. However, our forced photometry method allows us to identify X-ray sources that may fall below the typical significance level cut and helps to mitigate the effects of Eddington bias (see Gibson & Brandt 2012). Since we already know that there is an optical source at a given position, we can accept a lower level of significance and still expect almost no false positives from random background fluctuations. Assuming a false detection probability of $0.27 \%$ (a 3σ confidence limit), we expect only ∼2 false positives from our 583 sources (605 sources minus 23 sources with contaminants). This approach is standard for deep field Spitzer data, and WISE catalog construction (e.g., Lang et al. 2014). In addition, a similar implementation has recently been performed on the CDF-S/N by Xue et al. (2016).

We determine the flux for each source using the same prescription followed in Goulding et al. (2012). A count rate to flux conversion is applied assuming a power-law spectrum with a photon index of 1.9, and Galactic ${N}_{{\rm{H}}}$ $\sim 1\times {10}^{20}\ {\mathrm{cm}}^{-2}$ (Kalberla et al. 2005). We calculate the X-ray luminosity for each of our sources using the DEEP2 spectroscopic redshifts, and bring each X-ray luminosity to the rest frame by K-correcting the luminosities using a typical power law with ${\rm{\Gamma }}=1.9$ (Brandt & Hasinger 2005). Finally, we calculate the hardness ratios (HRs) for each of our sources, which is given by

$$\begin{eqnarray}&&\mathrm{HR}=\displaystyle \frac{H-S}{H+S},\end{eqnarray} \tag{ 1 }$$

where H is the number of HB counts and S is the number of SB counts. We use the Bayesian Estimation of Hardness Ratios code (Park et al. 2006) to calculate the HR and its associated uncertainties. HRs vary from −1 to 1, with typically more positive values for AGNs and more negative values for X-rays produced by stellar processes (Brandt & Hasinger 2005).

For each of our sources, we compute a false detection probability, which gives the probability of a spurious X-ray source being falsely identified as a significant source. In order to compute the false detection probability, we assume that the number of background counts have a Poisson distribution with a mean equal to the number of counts observed in the background region normalized to the area of the source region for each source. We then sample from this distribution 10⁶ times. The false detection probability is the percentage of times that the sampled distribution produced counts greater than the observed source counts. We calculate this probability for the SB, HB, and FB. The false detection probability is used to set the significance of our sources (i.e., a false detection probability $\lt 4.55 \%$ denotes a $\geqslant 2\sigma$ source). We consider a source to be a $\geqslant 3\sigma$ or $\geqslant 2\sigma$ detection if it is significant to that level in any of the three bands. Given the effective area of the ACIS instrument, Chandra is most sensitive and has the smallest PSF at soft energies, where the background is also the lowest. In the low-number Poisson regime, there is a further trade-off that more robust detections can be made over the widest possible energy band. Thus, our false detection probabilities are assigned as follows: FB, SB, and HB. Note that the NMBS and Chandra astrometry is matched within $\sim 0\buildrel{\prime\prime}\over{.} 9$ (Nandra et al. 2015), so we allow our $\geqslant 3\sigma$ X-ray centroid positions to vary by up to $1^{\prime\prime}$ before applying the counts to flux conversion.

We consider any $\geqslant 3\sigma$ sources (those with false detection probability $\lt 0.27 \%$) to be robust X-ray detections. These 10 sources are shown circled in blue in Figure 2 and their properties are given in Table 3. Their HST and Chandra X-ray Observatory images are given in Figure 3. Of the 10, seven are significant in the FB and one other band, two are only significant sources in the SB, one is only significant in the HB, and two sources are significant in all three bands. Note that the observed SB and HB correspond to rest-frame $0.75\mbox{--}3\,\mathrm{keV}$ and $3\mbox{--}10\,\mathrm{keV}$, respectively, at z = 0.5. The redshifts range from $z\approx 0.08\mbox{--}0.53$, though most of the sources are below $z\sim 0.3$. This could be due to the incompleteness of the DEEP2 spectra, which we discuss further in Section 4.2. The rest-frame X-ray luminosities of these sources range from ${L}_{{\rm{X}}}\sim 4\times {10}^{39}\ \mathrm{erg}\ {{\rm{s}}}^{-1}$ to $2\times {10}^{42}\ \mathrm{erg}\ {{\rm{s}}}^{-1}$. The median hardness ratio for these sources is −0.09 and the median SFR is $\sim 0.3\,{M}_{\odot }\ {\mathrm{yr}}^{-1}$. Goulding et al. (2012) performed a blind X-ray search in the AEGIS field, and found three sources that correspond to galaxies in the NMBS within our stellar mass range. Two of those sources, 17650 and 34347, are also identified by our method. Our estimated ${L}_{{\rm{X}}}$ agrees with that from Goulding et al. (2012) for both sources. The final source they identified, 31041, was removed from our sample because it is a merging system.

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Table 3.  Basic Properties of $\geqslant 3\sigma$ Sources

ID	${\alpha }_{{\rm{J}}2000}$	${\delta }_{{\rm{J}}2000}$	zspec	Stellar Mass	SFR	${L}_{{\rm{X}}}$ a	HRb	${P}_{\mathrm{False}}$ c	${P}_{\mathrm{XB}}$ d	[N ii]/Hαe	[O iii]/Hβf
#	deg	deg		$\mathrm{log}(M/{M}_{\odot })$	$\mathrm{log}({M}_{\odot }\ {\mathrm{yr}}^{-1})$	$\mathrm{log}(\mathrm{erg}\,{{\rm{s}}}^{-1})$
7290	214.37594	52.50924	0.1729	9.08	−0.40	${40.2}_{-0.2}^{+0.2}$	${0.96}_{-0.07}^{+0.04}$	9.01 × 10−4	0.0120	⋯	4.31g
14021	214.65990	52.64013	0.2083	9.30	−1.9	${40.8}_{-0.2}^{+0.1}$	$-{0.38}_{-0.2}^{+0.4}$	5.58 × 10−4	$\lt 1\times {10}^{-5}$	⋯	2.34
17650	214.51356	52.70632	0.1789	9.39	0.47	${40.7}_{-0.2}^{+0.1}$	$-{0.68}_{-0.1}^{+0.2}$	8.20 × 10−4	0.124	⋯	3.36
19957	214.76793	52.75079	0.3545	8.95	−0.35	${41.2}_{-0.2}^{+0.1}$	${0.90}_{-0.1}^{+0.1}$	1.27 × 10−3	$\lt 1\times {10}^{-5}$	⋯	3.71
31097h	214.50167	52.47253	0.5321	8.74	−0.73	${42.1}_{-0.1}^{+0.1}$	${0.38}_{-0.3}^{+0.3}$	<1 × 10−5	$\lt 1\times {10}^{-5}$	0.16	⋯
32364h	214.52787	52.59865	0.2036	9.42	−0.42	${40.5}_{-0.2}^{+0.1}$	$-{0.39}_{-0.3}^{+0.4}$	1.91 × 10−4	$1.68\times {10}^{-3}$	0.47	⋯
33817	214.79648	52.73136	0.08120	8.96	−0.60	${39.9}_{-0.2}^{+0.2}$	$-{0.13}_{-0.3}^{+0.5}$	1.99 × 10−3	0.0318	0.29	⋯
33915	214.53242	52.73998	0.06730	9.19	−1.2	${39.6}_{-0.3}^{+0.3}$	$-{0.79}_{-0.2}^{+0.2}$	<1 × 10−5	$3.47\times {10}^{-3}$	.21	⋯
34168h	214.82505	52.76636	0.3457	8.53	−1.0	${41.2}_{-0.1}^{+0.1}$	${0.11}_{-0.2}^{+0.3}$	<1 × 10−5	$\lt 1\times {10}^{-5}$	0.19	⋯
34347	214.82049	52.78238	0.4544	9.48	0.11	${41.4}_{-0.1}^{+0.1}$	$-{0.87}_{-0.1}^{+0.1}$	6.80 × 10−5	$6.77\times {10}^{-5}$	0.38	⋯

Notes.

^aX-ray luminosity, as calculated using the net counts in the FB (0.5–7 keV). ^bHardness ratios, as calculated using BEHR. ^cFalse detection probability, see Section 2.2 for a full description. ^dProbability the source is a HMXB, see Section 3.2 for a full description. ^e[N ii]/Hα line ratio from the DEEP2 data. ^f[O iii]/Hβ line ratio from the DEEP2 data. ^g[O iii] from 4959 Åline ([O iii]5007 Å = 3x[O iii]4959 Å). ^hPosition reflects X-ray source position rather than optical position.

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Using the HST data, we can glean some basic information about the morphologies of our sources. Most are late-type spirals at a variety of inclinations (see Figure 3). However, one (19957) is irregular, and one at high redshift (34347) is compact with little else about its morphology readily identifiable. A more quantitative analysis of the morphologies (e.g., to determine their bulge-to-disk ratios) is beyond the scope of this paper.

Our $\geqslant 2\sigma$ sources (those with false detection probability $\lt 4.55 \%$) are included as gray x's in Figures 4 and 5, and their properties are included in Tables 1 and 2. As described in Section 2.2, because our $\geqslant 2\sigma$ sources are at known galaxy positions, the vast majority are real sources. We include these sources because they enable us to have a large enough sample to give better statistics on the AGN properties of our sample as a whole (Section 4).

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There are multiple processes that may produce X-ray emission in a star-forming galaxy other than SMBH accretion. High-mass X-ray binaries (HMXBs) are the dominant source of X-rays from star-forming regions (e.g., Fragos et al. 2013). Hot gas associated with supernovae remnants can also emit in the X-ray, though this should be a very small factor for such low SFRs (see Mineo et al. 2012b). At ${L}_{{\rm{X}}}\gt {10}^{40}\ \mathrm{erg}\ {{\rm{s}}}^{-1}$, HMXBs contribute many orders of magnitude more to the luminosity than the low-mass X-ray binaries (see, for example, Lehmer et al. 2010; Mineo et al. 2012a). Thus, we will mostly consider the contribution from HMXBs, but will also discuss LMXBs and X-ray emitting hot gas from supernovae remnants as AGN contaminants.

Although we cannot measure detailed X-ray spectra with so few counts, we can use hardness ratios as a proxy for the type of X-ray emission we are observing. Obscured AGNs (intrinsic ${N}_{{\rm{H}}}\gtrsim 1\times {10}^{22}\ {\mathrm{cm}}^{-2}$) normally have the highest hardness ratios of $\mathrm{HR}\gt 0$, followed by unobscured AGNs with $\mathrm{HR}\sim -0.5$. HMXBs and X-ray emitting hot gas associated with supernovae remnants normally have very low hardness ratios of $\mathrm{HR}\leqslant -0.8$ (Brandt & Hasinger 2005). The hardness ratios for our $\geqslant 2\sigma$ and $\geqslant 3\sigma$ sources are given in Tables 2 and 3, and plotted versus the spectroscopic redshift in Figure 5. We plot the expected hardness ratio as a function of redshift for an obscured AGN with ${\rm{\Gamma }}=1.9$ and intrinsic ${N}_{{\rm{H}}}=3\times {10}^{22}\ {\mathrm{cm}}^{-2}$ (black, solid line), an unobscured AGN with ${\rm{\Gamma }}=1.9$ (blue, dashed line), an HMXB with ${\rm{\Gamma }}=3$ (green, dotted–dashed line), and hot gas, for which we use the APEC model provided by Sherpa, at ∼0.5 keV (red, dotted line). We calculate each of these with the PIMMS online calculator⁹ and allow for a constant galactic ${N}_{{\rm{H}}}=1\times {10}^{20}\ {\mathrm{cm}}^{-2}$. As shown in the figure, three of our sources (7290, 19957, and 31097) are clearly obscured AGNs because they fall well above the track shown for the model unobscured AGN. Most of our other sources are consistent with being either an obscured or unobscured AGN. Our three softest sources (17650, 33915, and 34347) appear inconsistent with either AGN model. These low HRs could be consistent with AGNs if these AGNs were extremely obscured (e.g., reflection dominated X-ray spectra produced in the presence of Compton thick absorption). On the other hand, these HRs are consistent with the normal X-ray emission from star formation. This second scenario seems more likely for two of these sources (17650 and 34347) considering their SFRs of $0.47\,{M}_{\odot }\ {\mathrm{yr}}^{-1}$ and $0.11\,{M}_{\odot }\ {\mathrm{yr}}^{-1}$, which are among the highest of our $\geqslant 3\sigma$ sources. Thus, based on HRs, we conclude that at least seven out of 10 sources are likely AGN powered.

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Now, we consider the possible contamination to the measured ${L}_{{\rm{X}}}$ due to HMXBs and LMXBs specifically. To quantify this contamination, we assign each source an X-ray binary (XRB) probability (i.e., the probability that the X-ray luminosity observed can be attributed to HMXBs or LMXBs). We assume Gaussian statistics, setting our mean to the observed X-ray luminosity in the FB,¹⁰ ${L}_{{\rm{X}}}$ and comparing it to the estimated X-ray luminosity due to XRBs, ${L}_{\mathrm{XRB}}$. We obtain ${L}_{\mathrm{XRB}}$ using the SFR given by the NMBS and the relation given by Lehmer et al. (2010):

$$\begin{eqnarray}&&{L}_{\mathrm{XRB}}=\alpha {M}_{\star }+\beta \mathrm{SFR},\end{eqnarray} \tag{ 2 }$$

where $\alpha =(9.05\pm 0.37)\times {10}^{28}\ \mathrm{erg}\,{{\rm{s}}}^{-1}\,{M}_{\odot }^{-1}$ and $\beta \,={(1.62\pm 0.22)\times {10}^{39}\mathrm{erg}{{\rm{s}}}^{-1}({M}_{\odot }{\mathrm{yr}}^{-1})}^{-1}$. This relation gives the luminosity in the $2\mbox{--}10\,\mathrm{keV}$ energy range at redshift z = 0. However, at the typical redshift of our sources, rest-frame 2–10 keV corresponds to observed 0.3–7 keV, which is approximately our FB. The predicted ${L}_{\mathrm{XRB}}$ for each of our sources is given in Table 1. The 68% confidence interval for ${L}_{\mathrm{XRB}}$ of each of our sources is shown in Figure 6. The colored points are the measured ${L}_{{\rm{X}}}$ in the FB for each of our $\geqslant 3\sigma$ sources, where the colors correspond to the redshift. The gray points give the measured ${L}_{{\rm{X}}}$ in the FB for all of our $\geqslant 2\sigma$ sources. Eight of our $\geqslant 3\sigma$ sources are more than $3\sigma$ away from the expected ${L}_{{\rm{X}}}$ from star formation.

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Since high-luminosity X-ray sources will only be populated stochastically at these SFRs $\lt \,1\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$ (Gilfanov et al. 2004), we also calculate the total number of luminous X-ray binaries expected in the entire sample, including any ultra-luminous X-ray sources (ULXs), which are X-ray sources with ${L}_{{\rm{X}}}\gt {10}^{39}\ \mathrm{erg}\ {{\rm{s}}}^{-1}$ associated with star-forming regions. We take the summed SFR for our galaxies with ${L}_{{\rm{X}}}\geqslant {10}^{40}\ \mathrm{erg}\ {{\rm{s}}}^{-1}$, and integrate the best-fit power law for the luminosity distribution of the HMXBs given by Mineo et al. (2012a) over the range of luminosities we measured. We find that we should expect $\lesssim 3$ HMXBs/ULXs with ${L}_{{\rm{X}}}\geqslant {10}^{40}\ \mathrm{erg}\ {{\rm{s}}}^{-1}$ in our entire sample. In line with our hardness ratio predictions, the distribution of ${L}_{{\rm{X}}}$ (given the SFRs in our sample) strongly suggests that at least seven out of 10 sources are powered by AGNs in the X-ray. We should stress that the X-ray binary luminosity function is still largely unknown—it is currently unclear how to include second order factors, like metallicity. There is some evidence that the X-ray binary luminosity function depends on metallicity, and is higher for metal-poor systems (Basu-Zych et al. 2016). However, this metallicity dependence is still not well known, so we do not attempt to correct for this effect here.

Another way of diagnosing the source of X-ray emission is to use the DEEP2 spectra to plot our sources on a Baldwin, Phillips & Terlevich (BPT) diagram (Baldwin et al. 1981). These diagrams use line ratios of strong optical emission lines to categorize the source of photoionization in galaxies. Normal star-forming galaxies occupy the left-hand locus of the BPT diagram (see Figure 8), with lower-metallicity sources typically having lower [N ii]/Hα and higher [O iii]/Hβ (e.g., Moustakas & Kennicutt 2006). AGNs appear in the upper-right of the diagram, while AGNs in dwarf galaxies, which also have low metallicity, tend to fall to the left of the typical AGN sequence due to their lower-than-average [N ii]/Hα (e.g., NGC4395; Groves et al. 2006; Ludwig et al. 2009).

The DEEP2 spectra for our $\geqslant 3\sigma$ sources are given in Figure 7. With the exception of 31097, which was at the exact redshift where all of the necessary emission lines were in detector gaps, all of our sources exhibit strong [N II], [O III], Hα, or Hβ emission lines. Due to the coverage of DEEP2, none of the spectra include all of the lines needed to properly place the sources in a BPT diagram. However, we can consider the sample as an ensemble by plotting the ratios we do have for each source (either [N ii]/Hα or [O iii]/Hβ). The intersection of these lines places the sample in a spot on the BPT diagram ([N ii]/Hα ∼ 0.15–0.35, [O iii]/Hβ ∼ 2–3) between normal SDSS dwarf galaxies and AGNs within SDSS (see Figure 8). This is consistent with the low-metallicity AGN described by Groves et al. (2006). Thus, the BPT diagram indirectly argues for AGN powering in our sources as well.

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Therefore, the combination of hardness ratios, ${L}_{{\rm{X}}}$/SFR, and optical line ratios together strongly imply that our sample is dominated by AGNs.

Eddington ratios are a useful tool for comparing AGN fractions from different studies; however, the masses of our black holes, ${M}_{\mathrm{BH}}$, are very uncertain. As mentioned in Section 1, we cannot measure the black-hole masses ${M}_{\mathrm{BH}}$ using dynamical methods. We also cannot use the ${M}_{\mathrm{BH}}\,-\,{\sigma }_{\star }$ relation to infer ${M}_{\mathrm{BH}}$ because we do not have ${\sigma }_{\star }$ measurements. In addition, the ${M}_{\mathrm{BH}}\,-\,{\sigma }_{\star }$ relation has not been well-measured in this low-mass regime (e.g., Barth et al. 2004; Dong et al. 2012), and may not hold for late-type spiral galaxies (e.g., Greene et al. 2010).

For lack of more detailed data, we simply assume that ${M}_{\mathrm{BH}}$ scales with M_⋆. This scaling is seen (e.g., Reines & Volonteri 2015), albeit with considerable scatter. To bracket this range, we take two extremes of the galaxy population at low mass: NGC 4395 and M32. NGC 4395 is a low-mass bulgeless spiral galaxy with a measured central SMBH mass of ${M}_{\mathrm{BH}}={4}_{-3}^{+8}\times {10}^{5}\,{M}_{\odot }$ (den Brok et al. 2015). M32 is a dwarf elliptical galaxy with a mass similar to that of NGC 4395, but with a measured central SMBH mass of ${M}_{\mathrm{BH}}=(2.4\pm 1.0)\times {10}^{6}{M}_{\odot }$ (van den Bosch & de Zeeuw 2010). We use NGC 4395 and M32 to bracket the range of ${M}_{\mathrm{BH}}/{M}_{\star }$. For NGC 4395, ${M}_{\mathrm{BH}}/{M}_{\star ,4395}=3.5\times {10}^{-4}$ (Reines & Volonteri 2015), while for M32 this is ${M}_{\mathrm{BH}}/{M}_{\star ,{\rm{M}}32}=8\times {10}^{-4}$ (van den Bosch & de Zeeuw 2010). We derive a corresponding average black-hole-mass range of ${M}_{\mathrm{BH}}\sim 5.5\times {10}^{5}\mbox{--}1.3\times {10}^{6}\,{M}_{\odot }$ for both our $\geqslant 2\sigma$ and $\geqslant 3\sigma$ sources. To find the Eddington ratios, we assume that ${L}_{{\rm{X}}}=0.1\times {L}_{\mathrm{bol}}$ (Marconi et al. 2004). The range of average Eddington ratios, ${L}_{\mathrm{bol}}/{L}_{\mathrm{Edd}}$, for our $\geqslant 3\sigma$ sources is 3%–7%, and 3%–6% for our $\geqslant 2\sigma$ sources.

This paper represents the first attempt to quantify the AGN fraction beyond $z\sim 0.4$ in $\sim {10}^{9}\,{M}_{\odot }$ galaxies. At the very least, the AGN fraction represents a lower limit on the fraction of dwarf galaxies that host massive black holes. As we will show, we are at the limits of the capabilities of both the optical and X-ray surveys, which makes quantifying our incompleteness challenging.

Let us start with our X-ray completeness. An examination of Figure 4 shows that for redshifts $0.2\lt z\lt 0.8$, we are incomplete below an X-ray luminosity of ${L}_{{\rm{X}}}\sim {10}^{41}\ \mathrm{erg}\ {{\rm{s}}}^{-1}$. Thus, for calculating our AGN fraction we will only consider sources with ${L}_{{\rm{X}}}$ greater than this limit across our redshift range. Our optical limit, on the other hand, has a more complicated selection function. Our stellar masses are derived from the NMBS, whose point-source detection limit is ${m}_{{\rm{K}}}=23.2\ \mathrm{AB}$. However, we also require a spectroscopic redshift from DEEP2, which introduces a known incompleteness for faint red galaxies with $z\gt 0.7$ (Newman et al. 2013). We examine the ratio of the NMBS to DEEP2 sources in redshift bins, and find that this ratio remains constant out to $z\sim 0.8$ if we restrict our attention to galaxies with ${M}_{\star }\gt {10}^{9}\,{M}_{\odot }$. However, to better compare with Aird et al. (2012; see Section 4.3), we we will only consider up to $z\lt 0.6$. Thus, for calculating an AGN fraction, we also restrict our attention to a mass and redshift range of ${10}^{9}\,{M}_{\odot }\leqslant {M}_{\star }\leqslant 3\times {10}^{9}\,{M}_{\odot }$ and $0.1\lt z\lt 0.6$.

The AGN fraction, f_AGN, for our sources within these limits is given by

$$\begin{eqnarray}&&{f}_{\mathrm{AGN}}=\displaystyle \sum _{i=0}^{N}\displaystyle \frac{(1-{P}_{\mathrm{false}}^{i})(1-{P}_{\mathrm{XRB}}^{i})}{{N}_{\mathrm{total}}},\end{eqnarray} \tag{ 3 }$$

where ${P}_{\mathrm{false}}$ is the false detection probability, ${P}_{\mathrm{XRB}}$ is the XRB probability (see Section 3.2), and${N}_{\mathrm{total}}=154$ is the total number of galaxies analyzed with mass ${10}^{9}\,{M}_{\odot }\leqslant {M}_{\star }\leqslant 3\,\times {10}^{9}\,{M}_{\odot }$ and $0.1\lt z\lt 0.6$. All of our $\geqslant 3\sigma$ sources with luminosity above the incomplete luminosity limit, ${L}_{{\rm{X}}}\,\gt {10}^{41}\ \mathrm{erg}\,{{\rm{s}}}^{-1}$, are included in this calculation. This leaves us with one $\geqslant 3\sigma$ source (34347), which would indicate an AGN fraction for our $\geqslant 3\sigma$ sources of 0.6%. However, because of the large uncertainties in this calculation, we can use the $\geqslant 2\sigma$ sources to provide an upper bound to the AGN fraction. In this case, we have five sources within the mass, redshift, and luminosity limits. This gives an AGN fraction for our $\geqslant 2\sigma$ sources of 3%. Thus, we find an AGN fraction of ∼0.6%–3%. This range of fractions is shown in Figure 9.

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Several studies have measured AGN fractions for dwarf galaxies in the local universe. It is highly non-trivial to compare our fractions with these other studies, given the different techniques, depths, and redshifts of the different surveys. In an attempt to provide somewhat meaningful comparisons, we here focus on X-ray surveys. Note that most of these studies cross-correlate optical positions of galaxies with an X-ray catalog rather than performing forced X-ray photometry at the optical positions as we do in this paper. Figure 9 and Table 4 give summaries of these AGN fractions.

Table 4.  Observed AGN Fractions

Study	Stellar Mass Range	Redshift Range	AGN Fraction	Parent Galaxy Selection Method
	$\mathrm{log}(M/{M}_{\odot })$
This work	$\lt 9.5$	0.1–0.6	0.6%–3%	NMBS
Aird et al. (2012) a	$\lt 9.5$	0.2–0.6	1.1%–4.5%	PRIMUS
Miller et al. (2015)	$\lt 10$	$\lesssim 0.008$	$\gt 20 \%$	Virgo + Field
Schramm et al. (2013)	$\lt 9.5$	$\lt 1$	$\sim 0.1 \%$	GEMS
Lemons et al. (2015)	$\lt 9.5$	$\lt 0.055$	$\sim 2 \%$	NASA-Sloan

Note.

^aExtrapolated from Aird et al. (2012), as discussed in Section 4.3.

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Miller et al. (2015) study ∼200 early-type dwarf galaxies in the local universe selected optically from HST imaging for the AGN Multi-wavelength Survey of Early-type galaxies (AMUSE) surveys (see Gallo et al. 2008; Miller et al. 2012, for details on these surveys). They then use the Chandra X-ray Observatory data from the AMUSE survey and Bayesian statistical techniques to study the AGN fraction in galaxies with ${M}_{\star }\lt {10}^{10}\,{M}_{\odot }$. Miller et al. (2015) report an SMBH occupation fraction of $\gt 20 \%$ down to $L/{L}_{\mathrm{Edd}}\sim {10}^{-4}$ for ${M}_{\star }\lt {10}^{10}\,{M}_{\odot }$. Due to our average sample redshift, we do not probe this low Eddington regime here. Lemons et al. (2015) also focus on the very nearby universe in their survey of dwarf galaxies. They select a sample of ∼44,000 dwarf galaxies from the NASA-Sloan atlas¹¹ with $z\lt 0.055$. Lemons et al. (2015) then cross-match these sources to the X-ray sources within the Chandra Source Catalog. Although they do not give the number of galaxies in the NASA-Sloan atlas that is within the Chandra Source Catalog area, we can estimate this number using the relative areas of each of these fields. The SDSS DR7 footprint is 11,663 deg² (Abazajian et al. 2009), and the Chandra Source Catalog SDSS observations have an area of $130\ {\deg }^{2}$ (Goulding et al. 2014). Thus, we estimate that of the ∼44,000 dwarf galaxies in the NASA-Sloan atlas Lemons et al. (2015) selected, ∼491 dwarf galaxies are within the Chandra Source Catalog area. Lemons et al. (2015) find 19 dwarf galaxies with hard-X-ray detections, though they argue that at most 10 of these are indicative of a central SMBH. This gives an estimated AGN fraction of $\sim 2 \%$ for ${M}_{\star }\lt 3\times {10}^{9}\,{M}_{\odot }$ and $z\lt 0.055$. At somewhat higher redshifts, Schramm et al. (2013) optically select ∼5200 dwarf galaxies with ${M}_{\star }\lt 3\times {10}^{9}\,{M}_{\odot }$ at $z\lt 1$ using HST/ACS imaging from the Galaxy Evolution from Morphologies and SEDs (GEMS) survey (Rix et al. 2004). They then use multi-wavelength catalogs to match these galaxies to X-ray sources in the Chandra Deep Field-South. Schramm et al. (2013) find 27 X-ray detected galaxies among their ∼5200 dwarf galaxy sample up to $z\lt 1$. However, they then focus on the sample with $z\lt 0.3$, optical spectra, and a high ratio of X-ray to optical luminosity, which leaves them with a parent sample of ∼2100 galaxies and 3 X-ray detected galaxies. They argue that the 3 galaxies in this sample all contain AGNs by examining their Hα or UV fluxes. Ignoring selection effects, we calculate an AGN fraction of $\sim 0.1 \%$. However, Schramm et al. (2013) does not discuss the incompleteness of the sample.

Perhaps the most straightforward comparison is made with the X-ray study of Aird et al. (2012). Aird et al. (2012) select 25,000 galaxies that are within fields with X-ray data from the PRIMUS survey (Aird et al. 2012). They identify 242 AGNs with ${L}_{{\rm{X}}}\gt {10}^{42}\ \mathrm{erg}\ {{\rm{s}}}^{-1}$ out of this sample using archival Chandra and XMM-Newton X-ray data. They report an AGN fraction of $\sim 1 \%$ up to $z\sim 1$ but for more massive galaxies than we consider here. While Aird et al. (2012) focus on different mass ranges than we do here, we can compare to their study because of the central result of that paper: Aird et al. (2012) show that AGN accretion rate distributions do not depend upon the host galaxy stellar mass. They report a universal Eddington ratio for ${M}_{\star }\gt 3\times {10}^{9}\,{M}_{\odot }$ and ${L}_{{\rm{X}}}\gt {10}^{42}\ \mathrm{erg}\ {{\rm{s}}}^{-1}$ in the redshift range¹²  of $z=0.2\mbox{--}0.6$. Given that the AGN accretion rate distribution is independent of the host galaxy stellar mass, we can extrapolate these probability densities to the mass and luminosity ranges we study here. For this linear extrapolation, we use Aird's fit to the figure of the active fraction (i.e., probability density of AGNs with a given luminosity, ${L}_{{\rm{X}}}$ in a galaxy of stellar mass, M_⋆) as a function of AGN X-ray luminosity and stellar mass (top panels of Figure 4 in Aird et al. 2012). By virtue of the universal accretion rate distribution, we adopt their measured slope (i.e., $P({L}_{{\rm{X}}}| {M}_{\star })$ per $\mathrm{log}{L}_{{\rm{X}}}/[\mathrm{erg}\,{{\rm{s}}}^{-1}]$) of $\alpha =-0.8$ (Aird et al. 2012). We then use a least squares fit to find the value for the AGN fraction at the "intercept" (${L}_{{\rm{X}}}={10}^{43}\ \mathrm{erg}\,{{\rm{s}}}^{-1}$) for ${M}_{\star }={10}^{9}\,{M}_{\odot }$, which gives us an intercept of $b=-3.3\pm 0.3$. Using these values, we calculate the extrapolated AGN fraction for $z=0.2\mbox{--}0.6$ and ${M}_{\star }={10}^{9}\,{M}_{\odot }$ (i.e., $P({L}_{{\rm{X}}}| {M}_{\star }={10}^{9}\,{M}_{\odot },z=0.2\mbox{--}0.6)$) to be 1.1%–4.5%. This is remarkably consistent with our own results of 0.6%–3% for ${10}^{9}\,{M}_{\odot }\leqslant {M}_{\star }\leqslant 3\times {10}^{9}\,{M}_{\odot }$ and $0.1\lt z\lt 0.6$ (see Figure 9).

We also compare the fractions of detected active black holes with the predictions of a semi-analytic model (SAM) of the joint formation and evolution of galaxies and SMBHs from Somerville et al. (2008; see also Hirschmann et al. 2012). SAMs are useful tools because they allow us to model a statistical sample down to very low masses, such as those relevant in this paper. The Somerville et al. (2008) models reproduce dwarf galaxy properties well, and manage to match the AGN luminosity function at higher luminosities (Somerville et al. 2012). Thus, it is interesting to ask whether their AGN fractions are similar to ours, with the eventual hope to test their assumptions about seeding and accretion.

The SAM is based on cosmological merger trees and contains recipes for the usual processes included in models of galaxy formation, including gas accretion and cooling, star formation, and stellar feedback. In this model, each top-level halo is seeded with a ${10}^{4}\,{M}_{\odot }$ black hole. As shown in Hirschmann et al. (2012), these seed masses are required to make the luminous AGNs observed at $z\gt 5$ without resorting to super-Eddington accretion. Black-hole accretion is triggered by mergers with mass ratios greater than 1:10, which is based on the results of hydrodynamic simulations of galaxy mergers (Hopkins et al. 2006, 2007). The black-hole growth is self-regulated by radiative feedback from the accreting black hole, resulting in a power-law decline of accretion rates as described in Hopkins et al. (2006). Although we consider a SAM in which black-hole accretion is triggered only by mergers, in reality, secular processes and internal disk instabilities likely contribute (see Hirschmann et al. 2012). While we exclude visible mergers from our survey, we show in Section 5 that including these mergers would not result in an appreciable change to our results. Moreover, many AGN hosts in the models would not be visually identified mergers. Many hosts at the accretion levels relevant here had mergers long enough ago that the signatures would no longer be easy to recognize, or were triggered by much smaller objects that would also be difficult to identify in surveys at the present depth.

We select galaxies from a mock AEGIS lightcone using the same stellar mass criteria and redshift range as for the observed sample. Our model predicts the accretion rate onto the black hole, and we convert this to an X-ray luminosity using a conversion of rest mass to energy of 0.1, and the bolometric correction to the hard-X-ray band from Marconi et al. (2004). We do not account for obscuration of the AGN radiation. We then compute the AGN fraction by counting the fraction of galaxies in the relevant stellar mass and redshift bins with ${L}_{{\rm{X}}}\geqslant {10}^{41}\ \mathrm{erg}\,{{\rm{s}}}^{-1}$.

In this way, we find a detected AGN fraction for galaxies with stellar mass of ${10}^{9}\,{M}_{\odot }\leqslant {M}_{\star }\leqslant 3\times {10}^{9}\,{M}_{\odot }$ and in the redshift range $0.1\leqslant z\leqslant 0.6$ of $3.0 \%$, which is encouragingly close to the observational results presented here (see Figure 9). The median Eddington ratio for the active black holes with ${L}_{{\rm{X}}}$ greater than the detection limit is 3%, while the mean for the same sample is 9%. We have also performed the exercise of adding the X-ray flux expected from HMXBs using Equation (2) at the given stellar masses and SFRs. We find that only a few additional galaxies would be above the adopted X-ray detection limit due to the added contribution to the X-ray flux from HMXBs, resulting in a negligible change in our predicted AGN fractions. This is consistent with the results shown in Figure 6.

While the SAM results agree surprisingly well with the observational results, it is important to understand that significant degeneracy still remains between the Eddington ratio distribution and the occupation fraction. In principle, the scenario that our SAM tests—that of high seed mass and overall low-accretion rates—may be correct, in accordance with direct gas cloud collapse models (e.g., Eisenstein & Loeb 1995; Regan & Haehnelt 2009; Volonteri 2010; Mayer et al. 2015) or formation in nuclear clusters (e.g., Loeb & Rasio 1994; Goswami et al. 2012; Katz et al. 2015; Stone et al. 2016). On the other hand, there may be a low seed mass and high accretion rates, which agree with Population III star collapse models (e.g., Madau & Rees 2001; Johnson et al. 2012; Becerra et al. 2015). Of course, the truth could be some hybrid of these multiple scenarios. However, the surprising agreement between our observations, the SAM, and the extrapolation from Aird et al. (2012) may suggest that the universal accretion rate distribution does extend to dwarf galaxies. If so, it is most likely that the occupation fraction is of the order unity (Miller et al. 2015; Plotkin et al. 2016). We are hopeful that more observational constraints, perhaps from JWST (e.g., Windhorst et al. 2009), will help inform these different scenarios. In addition, further studies similar to our own that focus on a wider mass range of galaxies would be beneficial in truly discerning the AGN fraction as a function of mass. This, combined with the observations from the next-generation telescopes, should allow us to constrain the seeding mechanisms further.

We also use the SAM to predict an AGN fraction for the same mass and X-ray luminosity limits in the redshift range $z=1\mbox{--}1.5$. We find an AGN fraction in this range of $9.4 \%$ and a mean Eddington ratio of $10.3 \%$ for the active black holes with X-ray luminosities above the detection limit. This predicted AGN fraction is a bit higher than the one we measure at lower redshifts, which could indicate a rise in AGN activity in dwarf galaxies similar to that in more massive galaxies at higher redshifts. This may be measurable in the near future by next-generation surveys.