D RAFT VERSION O CTOBER 24, 2019
Preprint typeset using LATEX style emulateapj v. 12/16/11

THE MASS RELATIONS BETWEEN SUPERMASSIVE BLACK HOLES AND THEIR HOST GALAXIES AT 1 < Z < 2
WITH HST-WFC3
X UHENG D ING 1, 2 , J OHN S ILVERMAN 3,4 , T OMMASO T REU 1 , A NDREAS S CHULZE 5 , M ALTE S CHRAMM 5 , S IMON B IRRER 1 , DAESEONGPARK 6 , K NUD JAHNKE 7 , VARDHA N. B ENNERT 8 , J EYHAN S. K ARTALTEPE 9 , A NTON M. KOEKEMOER 10 , M ATTHEW A. M ALKAN 1 ,DAVID S ANDERS 11
Draft version October 24, 2019

ABSTRACT
Correlations between the mass of a supermassive black hole (SMBH) and the properties of its host galaxy(e.g., total stellar mass M∗ , luminosity Lhost ) suggest an evolutionary connection. A powerful test of a coevolution scenario is to measure the relations MBH -Lhost and MBH -M∗ at high redshift and compare with local estimates. For this purpose, we acquired Hubble Space Telescope imaging with WFC3 of 32 X-ray-selectedbroad-line (type-1) AGN at 1.2 < z < 1.7 in deep survey fields. By applying state-of-the-art tools to decompose the HST images including available ACS data, we measured the host galaxy luminosity and stellar mass
along with other properties through the 2D model fitting. The black hole mass (MBH ) was determined using
the broad Hα line, detected in the near-infrared with Subaru/FMOS, which potentially minimizes systematiceffects using other indicators. We find that the observed ratio of MBH to total M∗ is 2.7× larger at z ∼ 1.5
than in the local universe, while the scatter is equivalent between the two epochs. A non-evolving mass ratio is
consistent with the data at the 2-3σ confidence level when accounting for selection effects (estimated using two
independent and complementary methods), and their uncertainties. The relationship between MBH and hostgalaxy total luminosity paints a similar picture. Therefore, our results cannot distinguish whether SMBHs andtheir total host stellar mass and luminosity proceed in lockstep or whether the growth of the former somewhat
overshoots the latter, given the uncertainties. Based on a statistical estimate of the bulge-to-total mass fraction,
the ratio MBH /M∗,bulge is offset from the local value by a factor of ∼ 7 which is significant even accounting
for selection effects. Taken together, these observations are consistent with a scenario in which stellar mass
is subsequently transferred from an angular momentum supported component of the galaxy to the pressure
supported one through secular processes or minor mergers at a faster rate than mass accretion onto the SMBH.
Subject headings: galaxies: active – galaxies: evolution
1. INTRODUCTION

Most galactic nuclei are thought to harbor a supermassive
black hole (SMBH), whose mass (MBH ) is known to correlate with the host properties, such as luminosity (Lhost ),stellar mass (M∗ ), and stellar velocity dispersion (σ∗ ). The
tightness of these correlations (also known as scaling relations) may indicate a connection between nuclear activity, andgalaxy formation and evolution (e.g., Magorrian et al. 1998;
Ferrarese & Merritt 2000; Marconi & Hunt 2003; Gültekindxh@astro.ucla.edu
1 Department of Physics and Astronomy, University of California, Los
Angeles, CA, 90095-1547, USA
2 School of Physics and Technology, Wuhan University, Wuhan 430072,
China
3 Kavli Institute for the Physics and Mathematics of the Universe (WPI),The University of Tokyo, Kashiwa, Chiba 277-8583, Japan
4 Department of Astronomy, School of Science, The University ofTokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan
5 National Astronomical Observatory of Japan, Mitaka, Tokyo 1818588, Japan
6 Korea Astronomy and Space Science Institute, Deajeon, 34055, Republic of Korea
7 Max-Planck-Institut für Astronomie, Königstuhl 17, D-69117, Heidelberg, Germany
8 Physics Department, California Polytechnic State University, San Luis
Obispo CA 93407, USA
9 School of Physics and Astronomy, Rochester Institute of Technology,
84 Lomb Memorial Drive, Rochester, NY 14623, USA
10 Space Telescope Science Institute, 3700 San Martin Drive, Baltimore,
MD 21218, USA
11 Institute for Astronomy, University of Hawaii, 2680 Woodlawn
Drive, Honolulu, HI 96822, USA

et al. 2009; Beifiori et al. 2012; Häring & Rix 2004; Gebhardt et al. 2001; Graham et al. 2011). Currently, the physical mechanism that can produce such a tight relationship is
unknown due to the daunting range of scales between the
dynamical sphere (∼pc) of the SMBH and their host galaxy(∼ 10 kpc). On one hand, cosmological simulations of structure formation are able to reproduce the mean local correlations, possibly by invoking active galactic nucleus (AGN)
feedback as the physical connection (Springel et al. 2005;
Hopkins et al. 2008; Di Matteo et al. 2008; DeGraf et al.
2015) or having them share a common gas supply (Cen 2015;
Menci et al. 2016), not necessarily in a direct manner. On theother hand, there may not be a need for a physical coupling(Peng 2007; Jahnke & Macciò 2011; Hirschmann et al. 2010),
the statistical convergence from galaxy assembly alone (i.e.,mergers) may reproduce the observed correlations.
To understand the nature of these correlations, it is important to study them as a function of redshift, determining how
and when they emerge and evolve over cosmic time (e.g.,
Treu et al. 2004; Salviander et al. 2006; Woo et al. 2006;
Jahnke et al. 2009; Schramm & Silverman 2013; Sun et al.
2015). During the past decade, there has been much progresson this front using type-1 AGNs. For example, it has been
demonstrated that AGN host galaxies, at z < 1 and for fixed
MBH , are under luminous compared to today’s hosts (Parket al. 2015; Treu et al. 2007a; Peng et al. 2006b). Similarly,
Bennert et al. (2011b) and Woo et al. (2008) found a positive
evolution of MBH , especially when the data are sufficiently
robust to isolate the luminosity or stellar mass of the bulge or

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X. Ding et al.

spheroidal component. At z > 1, Merloni et al. (2010) decomposed the entire spectral energy distributions (SED) into
a nuclear AGN and host-galaxy components and found a positive evolution of the mass ratios of black holes to their hostgalaxies. Although, the accuracy of such an approach for luminous AGNs has not yet been well established as discussed
herein. If realized, such offsets can be interpreted as a scenario in which SMBHs were built up first with galaxies thengrowing around their deep potential wells. A possible mechanism to account for the latter part of the growth of the galaxy
without increasing MBH is the transfer of stellar mass fromthe disk to the bulge (Jahnke et al. 2009; Bennert et al. 2011a;
Schramm & Silverman 2013) through bar instabilities or minor mergers.
However, there are studies (Cisternas et al. 2011; Schramm& Silverman 2013; Mechtley et al. 2016) based on HST imaging of deep survey fields such as COSMOS and CDFS thatreport no evolution in the MBH - M∗ mass ratio as compared to the local relation. In support, Sun et al. (2015) has
re-analyzed the mass ratios for broad-line AGNs in the COSMOS field in a similar manner (i.e., stellar mass measurements from SED fitting) to Merloni et al. (2010) and finds
no evolution when accounting for selection effects describedin Schulze & Wisotzki (2014) which consider the black holemass function and Eddington rate distribution of the sample
at their respective epoch.
To make substantial progress, it is important to constructstatistical high-z samples that reduce the uncertainties, andcarefully consider selection effects and inherent systematic
errors. First, one needs to deal with the inherent uncertainties in BH mass estimates using the so-called “virial” method.
In particular, many studies rely on MBH estimates using theCIV (or MgII ) line that may have unknown systematics, suchas a non-gravitational component of the gas dynamics of thebroad-line region (BLR), when compared to local samples
with masses based on broad Balmer lines (i.e., Hα and Hβ,Schulze et al. 2018; Baskin & Laor 2005; Trakhtenbrot & Netzer 2012). Second, measurements of the host galaxy properties are challenging due to the overwhelming glare of thebright nuclear light. This ultimately requires careful modelingof the point spread function and, whenever possible, the useof lensed AGNs since the magnification increases the spatial
resolution (Peng et al. 2006b; Ding et al. 2017a,b). Especiallynot to be overlooked as in past studies, the selection functionneeds to be taken into account when interpreting the observations (Treu et al. 2007b; Lauer et al. 2007). For instance, it has
been demonstrated by Schulze & Wisotzki (2011, 2014) that
selecting bright AGNs at high redshift results in a MBH - M∗relation with a steeper slope than if chosen randomly, suggesting the selection effects are the culprit rather than an intrinsically faster or even an existing evolution. It is also important
to consider the selection function when comparing observed
scaling relations with those from simulations (DeGraf et al.
2015).
In this study, we aim to make progress by utilizing a large
sample with high-quality data both for the measurement of the
MBH and their host properties, extending to higher redshifts
where evolutionary effects should be strongest. In particular, samples based on 2D image analysis using HST at z > 1
are limited. To date, the infrared capabilities of HST/WFC3
have not been fully exploited on this topic. Here, we measure the properties of 32 host galaxies with a redshift rangeof 1.2 < z < 1.7 using HST/WFC3 imaging data, and estimate their MBH based on the robust Hα detections, using

the multi-object spectrograph Subaru/FMOS. Given the highquality and sample size of our data, we are capable of testingwhether the growth of BH predates that of the host by a factor
of at least 1.7 (i.e., ∼ 0.23 dex, Schulze & Wisotzki 2014).
This value is the minimum offset expected for a non-evolving
mass ratio as described in Section 2.3. Furthermore, we collect from the literature comparison samples of intermediatez and local AGN, selected to have been analyzed with very
similar methods to those applied in the distant sample. We
recalibrate the relevant quantities from the literature based ona set of self-consistent recipes to ensure that our differential
measurement of evolution is robust to calibration and methodological issues (Section 2.4).
The paper is organized as follows. We describe the sample selection and their BH masses in Section 2. We describe
the new HST/WFC3 observations, available HST/ACS imaging, and construction of a PSF library in Section 3. In Section 4, we describe our method to decompose the rest-frame
optical emission and measure the host galaxy surface photometry. In Section 5, we use the multi-band host magnitudes toinfer their stellar population from which we apply to derive
the rest-frame R-band Lhost and M∗ to compare with local
relations. In addition, we use the information on the radial
light distribution (i.e., Sérsic index) to infer the likely fractionof stars in the bulge (M∗,bulge ) and its relation to the MBH .
The discussion and conclusions are presented in Section 6 andSection 7. Throughout this paper, we adopt a standard concordance cosmology with H0 = 70 km s−1 Mpc−1 , Ωm = 0.30,
and ΩΛ = 0.70. Magnitudes are given in the AB system. A
Chabrier initial mass function (IMF) is employed consistently.
2. EXPERIMENTAL DESIGN

We utilize a sample size of 32 broad-line (FWHM> 2000
km s−1 ; type-1) AGNs that have black hole mass measurements and fall within deep extragalactic survey fields that offer rich ancillary data. Specifically, we focused on meetingthe following criteria to overcome the limitations of previous
studies:• Black hole mass estimates (MBH ) are based on Balmerlines (i.e., Hα) which avoid potential systematic uncertainties in UV-based estimators (Greene & Ho 2005).
• Black holes masses MBH /M . 8.7, are below theknee of the black hole mass function to minimize selection biases (see Figure 1, top panel).
• Eddington ratios above 5% to further ensure homogeneity.
• X-ray selected sample has host-to-total flux ratios typically above 30% which facilitates the galaxy mass measurements.
• HST/WFC3 imaging of the host galaxy at rest-frame
wavelength ∼ 5500 Å, which is above the 4000 Å breakand does not include the broad Hα line (6563 Å), which
ensures sensitivity to the total stellar mass content. Thisis somewhat coupled to the previous item in this list.
• A large fraction of the sample has additional HST imaging (i.e., ACS), providing color information to achieve
reliable K-corrections and stellar mass determinations.

Cosmic evolution of the MBH -host relationlog (MBH /M ) . 8.5. The bolometric luminosities are measured from the broad AGN lines by Schulze et al. (2018, Section 3.3), which are used to calculate their Eddington ratio
λ = Lbol /LEdd . The flux-limited nature of the sample results in a slightly higher Eddington ratio distribution at lower
MBH as shown in Figure 1. Moreover, we have a higher preference to select targets which have rest-frame UV images (i.e.,
HST/ACS) as provided by Scoville et al. (2007) and Koekemoer et al. (2007) in the COSMOS field. We list the 32 AGNs
observed with HST/WFC3 and analyzed in this work in Table 1 sorted by field and redshift.

0
2
4
0.25
0.00

log(Lbol/LEdd)

3

0.25
0.50
0.75
1.00
1.25
1.50

COSMOS
Peng 2006a

1.75
7.5

8.0

SXDS
Decarli 2010

8.5

9.0

log(MBH/M )

CDFS

9.5

8 6 4 20log ( )F IG . 1.— Selection window used to choose our AGN sample based on
MBH and Eddington ratios (λ = LBol /LEdd ). As indicated by the vertical
red line, our sample (color coded) falls below the knee of BH mass function atz = 1.5 (top panel, Schulze et al. 2015). On the right, we illustrate the shapeof the Eddington rate distribution at these redshifts from the same reference.
For comparison, we plot the high-z luminous SDSS AGN samples that havebeen studied with HST (grey squares and circles from Peng et al. 2006a;
Decarli et al. 2010, respectively), mainly at the high mass end.

2.1. Sample selection
The 32 AGNs are initially selected by the X-ray observations of COSMOS (Civano et al. 2016), (E)-CDFS-S (Lehmer
et al. 2005; Xue et al. 2011), and SXDS (Ueda et al. 2008)
fields. In most cases, the X-ray sources are first identified as broad-line AGNs through optical spectroscopic campaigns with the VLT, Keck, and Magellan. Follow-up nearinfrared spectroscopic observations of the AGNs in thesefields are carried out with Subaru’s Fiber Multi-Object Spectrograph (FMOS, Kimura et al. 2010; Nobuta et al. 2012;
Matsuoka et al. 2013), covering the wavelength range 0.9 −1.8 µm which provides the favorable Hα and Hβ lines out
to z ∼ 1.7 to estimates the MBH . Recently, Schulze et al.
(2018) present near-IR spectroscopy of a large compilation of
243 X-ray AGN in these fields. It is from this catalog that
we primarily select our targets. The continuum fitting andemission-line modeling provided by this work are performed
through a common procedure for spectral model fits. Thisapproach first corrects the spectra for galactic extinction andshifts them to the rest-frame. The spectral regions in the nearIR that are strongly affected by OH emission are masked. The
spectrum is modeled by both a narrow-line and a broad-line
AGN template, and the emission line information is inferred
based on χ2 minimization. We refer the reader to the aforementioned paper for further details. Note that three CDFS
objects (i.e., CDFS-1, CDFS-229 and CDFS-724) are not included in Schulze et al. (2018); their infrared spectra are provided by Suh et al. (2015) using a similar approach.
Based on the MBH estimates described below (Section 2.2), we select targets with masses between 7.5 .

2.2. Details on BH mass estimates
The MBH of type-1 AGNs can be determined using the socalled virial method (Peterson et al. 2004; Shen 2013). The
kinematics of the BLR trace the gravitational field of the central supermassive black hole, assuming the gravity dominates
the motion of the BLR gas. Under these assumptions, thewidth of the emission-line provides the scale of the velocity
dispersion (∆V ), while the AGN continuum luminosity establishes an empirical scale of the BLR size (RBLR ). The estimation of MBH is then achieved using these measurements,
i.e., MBH ' G−1 RBLR ∆V 2 (McLure & Dunlop 2004).
To avoid any systematic bias between samples in the literature, we adopt a class of self-consistent estimators for ouranalysis. We first compare the estimators implemented in
Schulze et al. (2018) and Ding et al. (2017b), and find very
consistent Hβ(FWHM(5100))-based masses (MBH r.m.s. <0.03 dex). However, there is a ∼ 0.2 dex inconsistency intheir Hα mass estimates. Therefore, we utilize the AGNs
in Schulze et al. (2018) that have both Hα and Hβ lines (35AGNs in total) and carry out a cross-calibration to determine
which Hα estimator has the best agreement between the twolines. As a result, the Hα estimator in Schulze et al. (2018)has better agreement with Hβ. Thus, we adopt the scheme
given in Schulze et al. (2018) for all AGN samples used in thisstudy, including the comparison samples described below.
log

MBH (Hα)
M






LHα
= 6.71 + 0.48 log
1042 erg s−1
FWHM(Hα)+ 2.12 log,(1)1000 km s−1

MBH (Hβ)
M





and

log


Lλ5100= 6.91 + 0.50 log
1044 erg s−1
FWHM(5100)+ 2.0 log.
(2)1000 km s−1

Note that these recipes are first provided by Vestergaard &Peterson (2006). Using these recipes, we estimate MBH byadopting the emission-line properties as measured by Schulze
et al. (2018) for all 32 AGNs. While fourteen AGNs haveemission line properties for both Hα and Hβ, we adopt thevalue of MBH based on the Hα emission line to have consistency across the sample. While most recipes are calibrated
against Hβ the line is typically weaker than Hα, hence lower
signal-to-noise. We provide the MBH measurements together
with the properties of the emission lines in Table 2.
It is worth noting that the absolute flux calibration of the
FMOS spectra is set to match the available ground-based IR

4

X. Ding et al.

imaging, UltraVISTA in the case of COSMOS. While an initial flux calibration is performed during the reduction of the
FMOS data using calibration stars, there can be differential
flux loss due to aperture effects, variable seeing conditions,and minor alignment issues with the instrument. The flux
normalization is effectively an aperture correction in the J or
H band depending on the source redshift. We note that thisprocedure does not correct for intrinsic variability that may
induce an additional error of 0.2 mag. We refer the reader to
Section 2.2 of Schulze et al. (2018) for full details on the flux
calibration.
2.3. Expected bias from the selection function
Our AGN sample is primarily selected based on the valueof MBH and Eddington ratio. It is well known that sample selection effects must be taken into account in order to interpretthe observed black hole-host relations and measure its evolution with redshift thus avoiding biases (Treu et al. 2007a;
Schulze & Wisotzki 2011; Bennert et al. 2011a; Schulze &Wisotzki 2014; Park et al. 2015). The main source of bias
is due to the fact that active samples are necessarily selectedbased on properties that correlate with black hole mass estimators (such as AGN luminosity, line strength, and width)
and thus one tends to favor overly massive black holes in presence of intrinsic scatter and observational errors. Correcting
for observational biases requires a well-characterized selection function, such as the one we have for our sample, and amodel of the black hole mass function, and the evolution of
the correlations between MBH , and other properties.
Given our selection function, we use the Bayesian framework introduced by Schulze & Wisotzki (2011, 2014) to estimate the expected bias. In this framework, under the assumption of no evolution of the correlations between MBH andhost-galaxy properties, one can compute the expected bias fora given sample, prior to the observations. The key ingredientof this model are the local MBH host-galaxy property correlations, the black hole mass function, and Eddington ratio distribution at the redshift of observation. The latter two quantities are estimated from the type-1 AGN distributions (Schulze
et al. 2015) and corrected to represent the parent populationof all active SMBH hosts.
Adopting our specific selection limits into the framework (i.e., log(MBH /M ) ∈ [7.5, 8.56], log(Lbol /L ) ∈[45.0, 46.2] and log(λ) ∈ [−2.0, 0.5]), we infer an expectedbias of +0.21 dex in the observed ∆ log(MBH ) for samples
at z ∼ 1.5, assuming the baseline choices for the inputs to themodel that include the local value of the mass MBH /M∗ ratio.
To reiterate, an offset in the observed mass relations of 0.2 dex
at z ∼ 1.5 can be considered consistent with no evolution in
the mass ratio. This bias correction should be considered an
approximate estimate, with an uncertainty that depends on theuncertainty of the inputs. Furthermore, if the scaling relations
actually evolve, one needs to introduce a model for the evolution in order to infer the underlying bias-corrected trends. We
will revisit these issues in Section 5.5.
2.4. Comparison samples
We make use of the MBH -Lhost and MBH -M∗ relationsin the literature for comparison with our high-z data. For
the local relation, we use the measurements of Bennert et al.
(2010) and Bennert et al. (2011a) (hereafter, B10 and B11)
to define our zero-point for local AGN samples. The sample by B10 consists of 19 AGN with the MBH -Lhost rela-tion determined with reliable MBH masses using reverberation mapping with an uncertainty level ∼ 0.15 dex. Note that
B10 only provides the single galaxy V-band luminosity. Ding
et al. (2017b) derive the galaxy R-band luminosity based on
the same early-type galaxy template spectrum adopted by B10
using K-correction (see Section 3.2 therein). The work of B11
contains 25 local active AGNs, where the MBH are measured
using the single-epoch method (MBH uncertainty level ∼ 0.4
dex). To track the local M∗ and MBH relations to higher values, we include 30 inactive galaxies (mainly ellipticals or S0)
from Häring & Rix (2004) (hereafter HR04). It is worth noting that the local inactive sample is mainly bulge dominated,
and we adopt the bulge mass for the entire local sample. In
other words, our local comparison is the MBH -M∗,bulge andnot those involving total quantities.
We include in our analysis published samples at intermediate redshifts to understand the evolution of these correlations. We select samples that were analyzed by members of
our team to ensure uniform measurements. The intermediate redshift AGNs that we include are 52 objects published
by Park et al. (2015) using a single band which are applicable for the MBH -Lhost relation at 0.36 < z < 0.57 and27 objects published by Bennert et al. (2011b) and Schramm& Silverman (2013) at 0.5 < z < 1.9. Similar to the B10
sample, the R-band luminosities of these intermediate redshiftsystems (79 in total) are obtained by Ding et al. (2017b) using K-corrections from the V-band based on the same stellar
template. Moreover, Bennert et al. (2011b) and Schramm &Silverman (2013) estimate the stellar masses of their 27 objects using multi-band imaging data, and we adopt them as
the MBH -M∗ comparison sample. In addition, we adopt asample of 32 MBH -M∗ measurements at 0.3 < z < 0.9 byCisternas et al. (2011) to compare with our sample. Across theintermediate redshift sample, we recalibrate the MBH using
the self-consistent recipes introduced in Section 2.2 including those estimated from MgII using the recipe in Ding et al.
(2017b).
The values for the all the MBH -Lhost comparison samples
are listed in the paper by Ding et al. (2017b) (Table 1 andTable 2 therein). The MBH -M∗ comparison samples are collected and summarized in the Appendix C, Table 7.
3. HST OBSERVATIONSHigh spatial resolution imaging is required for the decomposition of the nuclear and host emission to accurately estimate the luminosity and stellar mass of the host galaxy. For
this purpose, we observed the sample of 32 AGNs, as described above, with the HST/WFC3 infrared channel, throughthe HST program GO-15115 (PI: John Silverman). We selected to use the filters F125W (1.2 < z < 1.44) and F140W(1.44 < z < 1.7) according to the redshift of the targets sothat the rest-frame spectral window is well above the 4000 Å
break. This selection further ensures that the broad Hα line is
not present in the bandpass so as not to contaminate the hostemissions due to the broad wings of the PSF.
For each target, we obtained six separate exposures of 399s(i.e., total exposure time 2394s). The six exposures were
dithered and combined with the ASTRODRIZZLE software
package following standard procedures, and resulted in an
output pixel scale of 0.00 0642 by setting pixfrac parameter as 0.8 and using a gaussian kernel12 . In Table 1, we list
12 For CID255, 3/6 of the dither WFC3 images are corrupted. We achieve
to analyze this sample using the same approach, taking the 3 available frames.

Cosmic evolution of the MBH -host relation5

the details of the individual observations.
Having obtained the HST image, we remove the background light, arising from the sky and the detector. In thisstep, we adopt the PHOTUTILS by Python and model the
global background light in 2D based on the SExtractor
algorithm, which effectively accounts for gradient in the background distribution. Then, we subtract the derived sky background light to obtain a clear image. To test the fidelity of thissubtraction, we measure the surface brightness in the empty
regions and verify that it is consistent with zero within thenoise. Finally, we extract the postage stamp of the AGN image and PSF image to carry out the modeling process; see
Section 3.1 and Section 4.
Multi-band information provides the SED at a more precise
level. A substantial fraction (21/32) of our objects have restframe UV images for those in COSMOS (Koekemoer et al.
2007). Here, we utilize images taken with ACS/F814W filter.
The final image is drizzled to 0.00 03 pixel scale. Given themulti-band images for our AGN, we are able to infer their
host color and assess the contribution of both the young andold stellar population which ensures an accurate inference ofrest-frame R-band luminosity (including a K-correction) andstellar mass (Gallazzi & Bell 2009).

point source and the host galaxy as a Sérsic profile. Note thatthe actual morphologies of the host galaxies could be more
complicated (e.g., bulge+disk). However, the Sérsic modelis an adequate first-order approximation of the surface brightness distribution with a flexible parameterization that provides
sufficient freedom to infer the total host flux even for our highredshift sample. We simultaneously fit the nearby galaxies,
that happen to be close enough to the AGN, with a Sérsic
model to account for any potential contamination from theirextended profiles. The systems CID206 and ECDFS-358 havenearby objects which could not be described by Sérsic model;
thus, we mask these objects in the fitting procedure.
We use the image modeling tool L ENSTRONOMY (Birrer
et al. 2015; Birrer & Amara 2018) to perform the decomposition of the host and nuclear light. L ENSTRONOMY is amulti-purpose open-source gravitational lens image forwardmodeling package written in Python. Its flexibility enables
us to turn off the lensing channel and focus on the AGN andhost decomposition13 . The main advantage of L ENSTRON OMY is that it returns the full posterior distribution of eachparameter (i.e., not just the best fit model) and the Laplaceapproximation of the uncertainties. The input ingredients to
L ENSTRONOMY include:

3.1. PSF library
The knowledge of the PSF is crucial for imaging decomposition of the AGN and its host, especially when the point
source contributes to the majority of the total emission. The
PSF is known to vary across the detector and over time due tothe effects of aberration and breathing. Simulated PSF, suchas those based on T INY T IM, are usually insufficient for our
purposes (Mechtley et al. 2012). Stars within the field-ofview of each observation provide a better description than thesimulated PSF since they are observed simultaneously withthe science targets, and reduced and analyzed in a consistentmanner Kim et al. (2008); Park et al. (2015). However, wehave found that such stars usually do not provide an ideal PSF
for de-blending the AGN and host galaxy through extensive
tests. The issues are associated with an insufficient numberof bright stars near our targets, color differences, and othereffects not fully understood.
To minimize the impact of such mismatches, we build a
PSF library by selecting all the isolated, unsaturated PSF-stars
with high S/N from our entire program. The selection consists of the following steps. First, we identify stars from the
COSMOS2015 catalog (Laigle et al. 2016). However, manybright stars with an intensity similar to our AGN sample were
excluded in this catalog. Therefore, we also manually select
PSF-like objects as candidates from the HST/WFC3 imaging.
We then discard non-ideal PSF candidates based on their intensity, FWHM, central symmetry, and presence of nearby
contaminants. In total, the PSF library contains 78 and 37
stars imaged through filters F140W and F125W, respectively.
We assume that the stars in the library are representative of thepossible PSFs in our program. The dispersion of PSF shapes
within our library provides us with a good representation of
the level of uncertainty in our measurements resulting fromthe image decomposition.

1. AGN imaging data.
– Using aperture photometry, we find that an aperturesize with radius ∼ 1.00 5 sufficiently covers the AGN
emission of our sample14 . By default, we extract animage of 61 × 61 pixels (i.e., 400 × 400 ). If needed, alarger box size is selected to include nearby objects.

4. AGN-HOST DECOMPOSITION

We simultaneously fit the two-dimensional flux distributionof the central AGN and the underlying host galaxy. Following common practice, we model the central AGN as a scaled

2. Noise level map.
– The origin of the noise in each pixel stems fromthe read noise, background noise, and Poisson noisefrom the astronomical sources themselves. We measure these directly from the empty regions of the data.
We then calculate the effective exposure time of eachpixel based on the drizzled WHT array maps to infer the
Poisson noise level. A final noise map includes all ofthese sources of error.
3. PSF.
– The PSF is directly taken from the PSF library. Usually, a mismatch exists when subtracting the AGN as
the scaled PSF, especially at the central parts. Whilemodeling multiply imaged AGN, this mismatch canbe mitigated with PSF reconstruction by the iterative
method (Chen et al. 2016; Birrer et al. 2018). However, this approach requires multiple images which are
not available in our case. We remedy this deficiency byusing a broad library which should contain sufficient
information to cover all the possible PSF.
The host property of an AGN is determined by the following steps. First, we model the AGN and host using each
PSF in the library. With the input ingredients to L ENSTRON OMY , the posterior distribution of the parameter space is cal13 As a check, we compared the results from L ENSTRONOMY to the commonly used galaxy modeling software G ALFIT and confirmed that the results
are consistent between the two.
14 The photometric aperture was determined as a compromise between thedetection of the total flux and a minimization of the background noise while
keeping the computational time low. We found that extending the radius beyond 1.00 5 did not add any missing light, while it increased the noise andcomputational time.

X. Ding et al.
60
50

data

model

data - Point Source2

normalized residual

4

pixel
6

10

15 20

30

data
model
QSO
host+1objs

CID1174

40

25
26(mag, pixel 2)

6

27

30

28

20

29

10
0

1"

1"

10 5 10 4 10 3 10 2 10 1 100

1"

10 5 10 4 10 3 10 2 10 1 10030

1"

10 5 10 4 10 3 10 2 10 1 100

6

4

2

0

2

4

6

0.1

0.2

0.5

arcsec1.0

2.0

F IG . 2.— AGN-host galaxy decomposition of COSMOS-CID1174 based on the HST/WFC3 F140W image.The panels from left to right are as follows: (1)observed data, (2) best-fit model (AGN+host), (3) data minus the model PSF (i.e., host galaxy free of the AGN), (4) residuals divided by the variance and (5) 1-D
surface brightness profiles (top) and the corresponding residual (bottom). The 1-D profiles indicate the surface brightness including the data (open circles), thebest-fit model (blue line), the AGN (orange line), and the model for the extended sources (green line, i.e., host and other objects). Note that the 1-D surface
brightness profiles are only for illustration purposes. The actual fitting is based on the two-dimensional images. The remaining 31 objects in the sample areshown in Appendix A.

culated and optimized by adopting the Particle Swarm Optimizer (PSO)15 (Kennedy & Eberhart 1995). To avoid any
unphysical results, we set the upper and lower limits on theparameters: effective radius Reff ∈ [0.00 1, 1.00 0], Sérsic indexn ∈ [0.3, 7]. Then, we rank the performance of each PSF
based on the χ2 value and select the top-eight PSFs as representative of the best-fit PSFs. We determine the host Sérsic
parameters (i.e., flux, Reff , Sérsic index) using a weighted
arithmetic mean, calculated as follows:
wi = exp − α(χ2i − χ2best ) ,2χ2best(3)

where the α is an inflation parameter16 so that when i = 8:αχ2i=8 − χ2best= 2.
2χ2best(4)

The goal of this recipe is to weight each PSF based on theirrelative goodness of fit, while ensuring at least eight are used
to capture the range of systematic uncertainties. The results
do not change significantly if we chose a different number of
PSFs, as shown below.
Note that since each AGN was observed at a different location of the detector and at a different time, the top-eight PSFs
usually vary from one AGN to another. Given the weights,
the value of host properties and the root-mean-square (σ) error are calculated as:
PN
xi ∗ w i,(5)x̄ = i=1
Σwi
s
σ=

PN

i=1 (xi − x̄)

Σwi

2∗w
i

,

(6)

where N is the number of the ranking PSF, i.e., N = 8. In
15 Note that L ENSTRONOMY enables one to further infer the parametricconfidence interval using MCMC. In our case, given a fixed PSF, the 1σ
inference of each parameter is extremely narrow. Thus, we only take thebest-fit inference using PSO for further calculations. The errors on the fit
parameters are assessed by using different PSFs for each object in the sample.
16 Defining α as the inflation parameter literally means that it has to belarger than 1. If α < 1, the relative likelihood between different PSFs would
be too close and introducing α would have a side-effect.

Figure 2, we demonstrate the best-fit result for COSMOSCID1174. The adopted weights are listed in Table 3.
We apply this approach to all AGNs to determine the global
characteristics of the hosts of type-1 AGNs at these high redshifts. We measure the effective radius (Reff ), Sérsic index,
and host-to-total flux ratio and describe each of these in thefollowing section. We recognize that these measurements areweighted by the eight top-ranked PSFs. The limited numberof top-ranked PSFs may underestimate actual uncertainties.
To gauge how the number of top-ranked PSF affects our results, we compare results when also using five and ten topranked PSFs. As shown in Figure 3, the results are consistent.
For each AGN, we check the location on the detector for the
top-ranked PSFs and find that they are unrelated to the AGN
position on the detector. This finding underscores that position on the detector is not the main factor driving the PSF
shape. Other factors, likely to be more significant, are thesub-pixel centering, the intrinsic color of the star, and the jitter and thermal status of the telescope during the observations.
The complexity of the problem highlights the necessity of decomposing the AGN using all the available PSF-stars from theentire program.
We carry out a similar analysis for 21/32 AGNs in the COSMOS field which have ACS/F184W imaging data. We assess
their host flux ratio using the same approach as done for the
WFC3-IR. The ACS FOV is more extensive than WFC3 one;thus we generated 174 PSFs for use. As expected, the detection of the host galaxy in the IR band is of higher significance than the UV due to the effects of dust extinction andthe contrast between the (blue) AGN and (red) host. Thus, we
fix the Reff and Sérsic n as the value determined by IR band
thus amounting to an assumption that the morphology of thegalaxy is consistent between ACS and WFC3 bands. In thiscase, the only free parameters are the total host and AGN flux.
We report the host galaxy properties in Table 4.
5. RESULTS

From our image decomposition, we detect the host galaxy
in all cases at a significant level, except one case (i.e., SXDSX763) that has a host-to-total flux ratio lower than 15%.
HST/WFC3 images with the AGN component removed arepresented in the third panel (i.e., the “data−point source”
stamps) of Figure 2 and in Appendix A for the remainingcases. While some of the galaxies have nearby neighbors,most are isolated and do not show strong signs of interaction

0.25
0.00
0.25
0.50

Cosmic evolution of the MBH -host relation

8

Rank 5 PSFs
Rank 8 PSFs
Rank 10 PSFs

6
#

4
2
0

20

40
60
Host flux ratio (%)

80
Rank 5 PSFs
Rank 8 PSFs
Rank 10 PSFs

8

#

6
4
2
0

0.2

0.4
0.6
effective radius (arcsec)

0.8

F IG . 3.— Distribution of the host-to-total flux ratio and effective radius
when implementing a different number of top-ranked PSFs.

or ongoing mergers, which indicates the fueling mechanismof AGNs may not be from major mergers for our sample. Thisis also relevant for model fitting with smooth Sérsic profiles
and the subsequent determination of the stellar mass. In thefollowing subsections, we describe the properties of our ensemble of type-1 AGN host galaxies.
5.1. Host galaxy properties
As shown in Figure 3 (top panel), the host-to-total flux ratio (total=host+nuclear) of the sample spans a wide range
from 10% to 90% with most of the sample concentrated between 20% to 50% (median value 37%). Those with flux ratios above ∼ 20% have a higher degree of significance withrespect to the detection of the host galaxy, while the fivesystems (i.e., CID255, CID50, LID360, CDFS-229, SXDSX763) which have host-to-total flux ratios lower than 20%
should be considered as marginal detections.
The distributions of effective radius Reff are shown in Figure 3 (bottom panel). Based on the redshift, we calculate thephysical scale of the radius for each object in kpc assuming
a standard cosmology. We plot them together with measuredSérsic index in Figure 4. These values are distributed within
the allowed range and not concentrated on either the upper orlower bound. The Reff of our sample are between ∼ 1−7 kpc(peaked at ∼ 2.2 kpc). Nearly half (15/32) of our systems
have Sérsic index n < 2, indicating they have a significant
disk component, likely in addition to the presence of a bulge.
Five systems have a high Sérsic index (n > 4.5) for the firstrun; we use n ∈ [1, 4] as prior to re-fit these systems and findthat the changes on the inference of their host luminosity arevery limited (< 0.03 dex). Seven systems have effective ra-7dius Reff < 0.00 18, i.e., smaller than 3 pixels. In order to check
whether this affects our conclusions, we re-fit them with aReff ∈ [0.00 2, 1.00 0] prior and find that the inferred host flux
barely changes (< 1%). In particular, the inferred Reff forthree systems (CID543, XID2202, and SXDS-X1136) have
hit the lower limit (i.e., 0.00 1), and thus the scatter on the parameter is formally zero. This means the size inference of
these three systems should really be considered an upper limit,
rather than a measurement. However, the inference of their
host luminosities and stellar masses are still reliable given theconsistency of the re-fitting results and the small scatter in thehost flux (< 10%).
We compare the morphology of AGNs host galaxies to inactive galaxies from the CANDELS imaging survey. We identify 4401 inactive galaxies within a redshift range (1.2 <z < 1.7), comparable to our AGN sample, whose Sérsic
measurements are provided by van der Wel et al. (2012) using G ALFIT. Their stellar masses are derived based on the
3-D-HST/CANDELS spectroscopic survey (Momcheva et al.
2016; Brammer et al. 2012) and comparable to our AGN hosts(9.5 < log(M∗ /M ) < 11.5; Section 5.4). We compare thehistogram of the inferred Reff and Sérsic index to the inactive
galaxies in Figure 4, where we find no significant difference
between their distributions and median value. We also test
whether the distributions can be drawn from the same parent
population using a Kolmogorov-Smirnov test and determining the p-value to be 0.42 and 0.04 for Reff and n, respectively. We conclude that the host galaxies of our AGN sample
are representative of the overall population of galaxies, witha significant disk component, at comparable luminosity andstellar mass at the same redshift. In Section 5.6, we use thisinformation to infer the likely bulge masses, hence the MBH– M∗,bulge relation.
5.2. Rest-frame colors
For 21/32 AGNs, we have multi-band host magnitudes
that enable us to select the appropriate stellar population
templates to determine rest-frame luminosities and stellar
masses for the overall sample. We find that the 1 Gyr andthe 0.625 Gyr stellar population with solar metallicity and a
Chabrier IMF (Bruzual & Charlot 2003) provide the excellent
matches to the observed colors of our sample at z < 1.44 andz > 1.44, respectively (see Figure 5). To minimize the uncertainty associated with these corrections, we use these two
templates to interpolate to the rest-frame R band which is very
close to the observed wavelengths. We note that the choice is
not unique, and other combinations of ages and metallicities
and star formation history could match the observed colors
and would provide very similar R-band magnitudes, and also
stellar masses (Bell & de Jong 2000, 2001).
5.3. MBH -Lhost relation
Adopting the 1 Gyr and 0.625 Gyr stellar population, we
perform a K-correction to derive the rest-frame R-band magnitude of our sample, based on the host inference in the WFC3
band. As mentioned above, since the WFC3 filter is alreadyclose to the rest-frame R-band, we expect the MR uncertainty
introduced by this K-correction to be within 0.05 mag. We derive the rest-frame R band luminosity from log LR /LR, =0.4×(MR, −MR ), where MR, = 4.61 (Blanton & Roweis2007). LR ranges between log(LR /L ,R ) ∈ [9.5, 11.5] withindividual values listed in Table 4.
We show the relation between MBH and Lhost in Figure 6

8

X. Ding et al.

0.454

1.50

stellar masses systematically). We find M∗ ranges betweenlog(M∗ /M ) ∈ [9.7, 11.3]. These values are listed in Table 4.
In Figure 7, we plot the our MBH – M∗ measurements andfind that the distributions in this plane for the four samples
are similar. However, there is a slight shift of the high-z
data towards higher black hole masses at a fixed host mass.
In Figure 8, we plot the mass ratio as a function of redshift(right panel) and the mass ratio difference (∆ log MBH =MBH
M∗
log 10
7 M ) − α − β log( 1010 M ) which is considered as thedifference between the high redshift data and the best-fit local
relation; left panel). For our high-z sample, we find the averaged ∆ log MBH to be 0.43 ± 0.06, a factor of ∼ 2.7 higher
than the local mass ratio as indicated by the top blue circle in
Figure 8.
As with Equation 7, we fit the MBH -M∗ data with a linear
relation as:

AGN sample
CANDELS sample

Number Density

1.25
1.00
0.75
0.50

0.336

0.25
0.00

0.5

0.0

0.4

0.5

1.0

log(Reff) (kpc)

2.0

AGN sample
CANDELS sample

1.403

Number Density

1.5

log

0.3

(8)

and an evolution term parameterized in the following form:

0.2
0.1
0.0

MBH
M∗
) = α1 + β1 log( 10
),
107 M
10 M

∆ log MBH = γ log(1 + z).

2.04
0

1

2

3

4

5

Sersic index

6

7

8

F IG . 4.— The comparison of the histogram of the Reff (top panel) andSérsic ns (bottom panel), with median value indicated. For comparison, weshow the distribution of CANDELS galaxies at similar redshifts and stellar
masses to our type-1 AGN sample.

with comparison samples. For reference, we fit the local datawith a linear relation,
log

MBH
LR
) = α0 + β0 log( 10
),
7
10 M
10 L

(7)

which enables a direct comparison between our high-z sample and the local relation. The distribution of our data appears to be in good agreement with the local relation and theother AGN samples at lower redshift. Therefore, the observational data indicate that the relation between black hole massand the host luminosity are similar at different periods of theuniverse. In Appendix B, we explore how the high-z sample
would evolve in this plane solely with the luminosity evolution of the host galaxy as done in past studies (e.g., Ding et al.
2017b).
We note that there are outliers that deviate from the distribution of the overall sample, such as SXDS-X763. We suspect that this source may have abnormal host properties. Indeed, the value of Reff has a large uncertainty (∼ 75%), andthe host-to-total flux ratio is the lowest of the high-z sample(< 10%).
5.4. MBH -M∗ relation
Using the NIR imaging with HST, we take the host luminosity along with color information to estimate the stellar mass content of each host galaxy based on the mass-tolight ratio of the adopted stellar populations. The uncertainty level associated with the stellar mass is expected to
be of order 0.1 dex (changing the IMF would affect all our(9)

Performing such a fit based on the observed data (i.e., no correction for selection effects), we obtain γ = 1.03±0.25. Even
with the exclusion of the few outliers, the slope is significantly
non-zero with γ = 0.89 ± 0.27. The best-fit observed evolution model is shown in Figure 8.
As shown in these panels, the scatter of the high-z correlation presents a similarity to the local relation. To quantitively
make this comparison, we investigate the intrinsic scatter ofour sample. Note that the observed high-z sample has effects from both the measurement uncertainty and the narrow
selection window, and thus one needs to take them both into
account to extract the intrinsic scatter. Based on our forward
modeling framework, we find that our AGN sample has intrinsic scatter as 0.25 dex on the vertical axis, taking into accountthe measurement uncertainty and the selection function. We
will return to this topic in a forthcoming paper (Ding et al. in
prep.), where we focus on the scatter of the sample, as a diagnostic of AGN-host galaxy feedback mechanism. The factthat the intrinsic scatter for the high-z sample is no larger thanthe local one, which is ∼ 0.35 dex (Gültekin et al. 2009), maypose a challenge for explanations of scaling relations where
random mergers are the origin of the correlation (Peng 2007;
Jahnke & Macciò 2011), indicating that there may likely be aconnection between the SMBHs and their host galaxies during their formation.
5.5. Taking into account the selection function
In Section 2.3, we used the Bayesian framework introducedby Schulze & Wisotzki (2011) and find that in the case of
no intrinsic evolution of the correlations we would expect tomeasure an offset corresponding to a +0.21 dex bias in the inferred ∆ logMBH , owing to our selection function. Interestingly, this bias could account for the majority of the observed
offset as obtained in the MBH -M∗ correlations in the last section. To demonstrate, we use blue open circles to show the
expected bias raised by this selection effect in Figure 8. We
also show, in Figure 9, the histogram ∆ logMBH of the highz sample and compare it to the local ones. The plots show

Cosmic evolution of the MBH -host relation

3.5

mag(ACS) - mag(WFC3)

3.0

Absolute rest-frame R mag by ACS

5 Gyr template
1 Gyr template
0.625 Gyr template

2.5
2.0
1.5
1.0
0.5

1.1

1.2

1.3

1.4

z

1.5

1.6

1.7

9

21.0
21.5
22.0
22.5
23.0
23.5
24.0

1.8

24.0

23.5

23.0

22.5

22.0

21.5

21.0

Absolute rest-frame R mag by WFC3

F IG . 5.— Left: The observed color as a function of redshift, based on 21/32 AGNs which have both WFC3 and ACS imaging data. Note that the filter
combinations WFC3/F125W and WFC3/F140W are adopted for galaxies at z < 1.44 and z > 1.44, respectively. We also plot three predicted models including
5 Gyr, 1 Gyr and 0.625 Gyr. Clearly, the 5 Gyr stellar population is ruled out by our data, and we adopt the 1 Gyr and 0.625 Gyr for galaxies at z < 1.44 andz > 1.44, respectively. Right: Comparison of the inferred rest-frame R band magnitude, based on the adopted stellar populations.

MBH Lhost relation

10.0

Local AGNs by Bennert+10
Intermediate redshift AGNs
This work

10.0

9.5

9.0

intermediate redshift AGNs
This work

9.0

sampleuncertainties

log(MBH/M )log(MBH/M )9.5 intermediate

MBH M * relation

Local by Bennert+11
Local by H&R

8.5
8.0
7.5

8.5
8.0
7.5

intermediate
sampleuncertainties

7.0

7.0
6.5

log(MBH/107M )=0.49+0.90log(LR/1010L )

9.0

9.5

10.0

10.5

11.0

log(LR, host/L )

11.5

6.5
12.0

F IG . 6.— The observed black hole mass vs. R-band luminosity relation.
The black line and the blue colored equation indicates the best-fit result of
the local sample as given in Equation 7, with 1σ confidence interval indicated by the gray region. The values for the comparison samples at low andintermediate redshift are listed in Ding et al. (2017b), Table 1 and 2.

that the correlations between MBH and host galaxy total stellar mass or luminosity could be consistent with those of thelocal samples given the uncertainties and selection function.
To further evaluate the “true” underlying evolution andits uncertainty, corrected for selection effects, taking into
account the actual observations, we adopt an independent
method based on the approach introduced by Treu et al.
(2007a) and developed by Bennert et al. (2010), Park et al.
(2015) and Ding et al. (2017b). The method parametrizes the

6.09.0log(MBH/107M )=0.27+0.98log(M * /1010M )
9.5

10.0

10.5

11.0

log(M * /M )

11.5

12.0

12.5

F IG . 7.— Black hole mass vs. stellar mass relation ( MBH – M∗ ). The
best-fit local relation is shown as described in Figure 6.

evolution of the correlations between MBH and host properties as an offset from the local one γ, and an intrinsic scatter σint , which can be a free parameter or tied to the local
relation. It then imposes a selection function in MBH andcalculates the intrinsic parameters of the model given the observations. In practice, we start from the local black holemass function and the evolution model, generate mock samples using a Monte Carlo approach, apply the selection function and compare with the data to generate the likelihood. To
illustrate the importance of the intrinsic scatter in the selec-

X. Ding et al.

0
0.5
3.5
Local by Bennert+11
3.0
Local by H&R
2.5
2.0
1.5
1.0
0.5
0.0
0.5
1.0
1.5
2.00.0
0.1
0.2

z 1

1.5

2

2.5

Intermediate redshift AGNs
This work

0

z 1

0.5
Local by Bennert+11
Local by H&R

10 1

1.5

2

2.5

Intermediate redshift AGNs
This work

MBH/M *

logMBH (vs M * )

10

10 2

10 3
0.3

log(1+z)

0.4

0.5

0.0

0.1

0.2

0.3

log(1+z)0.4

0.5

F IG . 8.— Left: Offset in log(MBH ) (vs. M∗ ) as a function of redshift. The orange band is the intrinsic scatter of local linear relation. The red line andgray band are the best-fit and 1σ offset fitting by Equation 9 using our 32 high-z AGNs, with slope value γ = 1.03 ± 0.25. Right: MBH /M∗ as a functionof redshift. In both panels, we use blue open circles to show the expected bias of the median value for ∆ logMBH . We use framework of Schulze & Wisotzki(2011, 2014) to show that even with no evolution selection effects would shift the expectation higher towards the measured values as indicated by the small arrow(Section 2.3.) The measurements of the local and intermediate redshift samples are re-calibrated using self-consistent recipes and listed in Table 7.

2.5

Local, Bennert+11
Local, H&R
High z sample

Number Density

2.0
1.5
1.0
0.5
0.0

0.21

1.0

0.5

0.0

0.41

0.5

log(MBH)

1.0

1.5

F IG . 9.— Histogram of the log(MBH ) offset for the local samples and our
32 high-z sample. The gray vertical line shows the mean value of the offset
for the high-z sample, and the red vertical line shows the shift the expectationby the selection effect, even if there were no evolution. The median value ofthe offset is very close to the mean value (+0.30).

tion effects, we adopt both uniform (flat) prior and lognormal
prior for σint . Note that this method assumes a narrow Eddington ratio distribution, which is different from the one by
Schulze & Wisotzki (2011), as described in Section 2.3. Also,
this method adopts the local black hole mass function rather
than the high-z BH mass function. These differences have asecond-order effect and could be responsible for the different magnitude of the selection effect. On the other hand, thismethod probes the importance of the scatter at high-z, whichis complementary.
Combining the 32 AGNs together with the intermediateredshift sample, we present the inferred γ and σint in the twodimensional planes in Figure 10. The plots show that the in-ferred evolution is uncertain and depends crucially on the intrinsic scatter, especially when applying a luminosity evolution for the host (see Appendix B). Assuming the lognormal
prior σint , to mimic the assumption of the method discussed
in Section 2.3, one sees that the best estimate of γ is positive,
but the 95% confidence intervals extend to zero. Thus one
cannot conclude that evolution is significantly detected in ourdata. When relaxing the prior on σint we find that the scatteris consistent with being as low as in the local samples, with aone-sided interval including zero and 1σ upper limit at around
0.5 dex. We also study the selection effect by only considering the new 32 AGNs, resulting in a higher evolutionary trendwith a higher value of γ. We show all the result of the γ andσint in Table 5.
A simple check using our prior estimate of the bias yields
a similar result. The mean offset of the sample of 32 objects
from the local relationship is 0.43 ± 0.06 dex, which would
correspond to γ = 1.08 ± 0.15. However, after correcting
for the selection bias (0.21 dex, see Section 2.3), the offset
reduces to 0.22 ± 0.06 and thus γ = 0.55 ± 0.15, marginally
positive, but not conclusively inconsistent with the local value
given the error bars and uncertainty in the correction. We also
note that the distribution of offsets shown in Figure 9 displays
a positive asymmetric tail. Whereas we expect the negative
tail to be suppressed by our selection function, and thus itshould be accounted for in our treatment. If one computes themedian offset instead of the mean, the offset is almost completely consistent with the expected bias. Larger samples withdifferent selection function are needed to establish whether
the positive tail is real or due to small sample statistics.
5.6. MBH -M∗,bulge relation
The local sample of inactive galaxies used in this analysis is mainly comprised of bulge-dominated galaxies, and theentire local M∗ we adopted are their bulge masses. That is,in previous sections, we are comparing the MBH -M∗,total
relations in the distant universe to the MBH -M∗,bulge relations locally. Considering that a significant stellar component

Cosmic evolution of the MBH -host relation(a) MBH -M∗ , flat prior

11(b) MBH -M∗ , lognormal prior

F IG . 10.— Constraining the evolution factor γ of MBH -M∗ relation (Equation 9), with intrinsic scatter σint , using a Monte Carlo simulation. The adopted
sample includes our 32 AGNs and the intermediate redshift AGNs, using flat prior of σint (left panel) and lognormal prior (right panel). The colored regions
indicate the 68 and 95% confidence regions.

1.0

0.8

B/T stellar mass ratioof the high-z AGNs have a disk component (i.e., AGN hosts
with fitted Sérsic index close to 1), M∗,bulge must be smaller
than M∗,total . Given that the structures of our AGN hosts are
similar to inactive galaxies at equivalent redshifts and stellar
masses (Section 5.1), we expect their B/T ratios to follow asimilar distribution that can be estimated using a single-Sérsic
index (Bruce et al. 2014). We can then infer the bulge stellar
mass of our AGN hosts based on the inferred B/T ratios.
First, we establish a relation between the single-Sérsic index and the B/T ratio using inactive galaxies from CANDELS at similar redshifts and stellar masses (Figure 11;log(M∗ /M ) ∈ [9.0, 11.5]) using the B/T measurements of
Dimauro et al. (2018). In the figure, the red line indicates theaverage B/T ratio at a given Sérsic index. We then implementa Monte Carlo approach by randomly sampling a GaussianSérsic distribution for each AGN in our sample based on our
measurements with 1σ errors (Table 4). Next, we randomly
sample the associated B/T ratio for a given Sérsic index. To
avoid the case of an unphysical faint bulge flux, we set thelower limit for B/T ratio as 0.1. In each realization, we compare the M∗,bulge to that of the MBH and estimate the offset
and γ. We use 10,000 realizations to make sure the distribution of random samples is stable. Note that the Monte Carloapproach would take the scatter into account, including theSérsic index and its relation with the B/T ratio. We list theresulting bulge masses M∗,bulge in Table 4.
As expected, we find a stronger offset of ∆ log Mbulge=
BH
0.87 ± 0.07 dex in MBH /M∗,bulge with respect to the local relation, which corresponds to an evolutionary trend with
γ = 2.09 ± 0.30. Since selection effects are independent of
B/T in our framework, we can apply our prior estimate of thecorrection for selection effects by removing 0.21 dex, leaving still a significant offset of 0.67 ± 0.07 dex. We show thehistogram of MBH /M∗ and γ inferred by the Monte Carloapproach in Figure 12 and illustrate the offset between bulge
and BH as a function of redshift in Figure 13.
Even with the uncertainty associated with our estimate of
the bulge-to-total ratio, it is clear that there is a significant
evolution in the MBH – M∗,bulge relation, considering themarginal evidence for evolution in the MBH -M∗,total relation and that our galaxies are not pure bulges. This finding
confirms with higher fidelity the conclusion by Bennert et al.
(2011b) based on optical data of a smaller sample of 11 ob-

0.6

0.4

0.2

The averaged B/T relation

0.0
0

1

2

3

4

5

Sersic index

6

7

8

F IG . 11.—
Relation between bulge-to-total stellar mass ratio
and single-Sérsic index using CANDELS inactive galaxies withlog(M∗ /M ) ∈ [9.0, 11.5]. The red curve is the averaged B/T ratio at different Sérsic index. Note that this curve is only for demonstrating
the expected B/T value at different Sérsic index. In this work we adopt theMonte Carlo approach to randomly sample the B/T to account for the scatter.

jects in the same redshift range (see Schramm & Silverman
2013; Jahnke et al. 2009; Cisternas et al. 2011, for similar results). We list the inferred γ using the different sample combinations in this section in Table 6.
6. FINAL TECHNICAL REMARKS

6.1. Systematic errors
In this work, we use state-of-the-art techniques to measure the host galaxy flux, separating it from that of the point
source. The fidelity of the host galaxy magnitude is high as
demonstrated by the robustness with respect to the choice of
PSF, which is the primary source of uncertainty. However, we12

X. Ding et al.

PDF

PDF

High-z sample
Local sample

10 3

10 2

10 1

101

100

MBH/M*, bulge

1.0

1.5

2.0

2.5

3.0

0
0.5
3.5
Local by Bennert+11
3.0
Local by H&R
2.5
2.0
1.5
1.0
0.5
0.0
0.5
1.0
1.5
2.00.0
0.1
0.2

z 1

1.5

2

2.5

0

Intermediate redshift AGNs
High-z bulge

z 1

0.5
Local by Bennert+11
Local by H&R

1.5

2

2.5

Intermediate redshift AGNs
High-z bulge

10 1

MBH/M*, bulge

logMBH (vs M*, bulge)

F IG . 12.— Illustration of the distribution from the 10, 000 time realizations. Left: MBH /M∗ distribution for 32 high-z sample, compared to the local one.
Right: γ distribution, with median value (blue line) and 1σ region (dashed line) .

10 2

10 3
0.3

log(1+z)

0.4

0.5

0.0

0.1

0.2

0.3

log(1+z)0.4

0.5

F IG . 13.— Same as Figure 8, but for MBH -M∗,bulge relations. The intermediate sample consists of 27 objects from Bennert et al. (2011b) and Schramm &Silverman (2013), listed in Table 7. Cisternas et al. (2011) do not provide information to infer their bulge mass.

introduce some uncertainty by adopting two common simple
stellar population templates to derive the rest-frame R-band
luminosity and stellar mass for the full population. In principle, if we had more information on the color, we could improve on this source of uncertainty by adopting a specific stellar population model for each target. However, consideringthat the host magnitude in the HST/ACS band is faint (hostto-total flux ratio < 30%), the individual color that we measure is insufficient to further discriminate between templates,
and thus we adopt the common templates for simplicity andto facilitate reproduction of our work.
In any case, we stress that the uncertainty in the scaling
relations is dominated by the single epoch black hole massestimates (i.e., 0.4 dex), and not by uncertainties in the photometry or stellar mass estimates (i.e., typically ∼ 0.15 dex).
Thus, the choice of template is a subdominant source of error. Note that the large uncertainties of MBH is not immediately apparent from the figures, since only the AGNs thathave measured MBH that fall into our selection window (i.e.,log(MBH /M ) ∈ [7.5, 8.56]) have been selected. A posteriori, the fact that the intrinsic scatter we observe is not larger
than the one in the local universe is consistent with this statement.
6.2. Systematic errors related to the choice of local anchor
To compare our high-z sample to the local relation, we
adopt the local measurements by Bennert et al. (2010, 2011a)and Häring & Rix (2004). In the literature (Kormendy & Ho
2013; Bentz & Manne-Nicholas 2018), other analyses of thelocal sample are available, which could also be consideredas the local anchor. However, from the point of view of ourdifferential evolutionary measurement, we stress once again
the importance of selecting local anchors that have been measured and calibrated in a similar way to the distant high-z
samples. It is crucial that the black hole masses be obtained in
the same of self-consistent manner, and it is also critical that

Cosmic evolution of the MBH -host relation
the host/AGN decomposition be measured in the most similar
way possible.
Therefore, for these reason, the B10 and HR04 samples
are the most appropriate for our goal, in spite of their limitations. For instance, B10 did not consider morphologicalfeatures like strong bars in the decomposition. This choicemight affect the inference of the bulge/disk properties in absolute terms, but it is the most similar decomposition to what
is possible at high-z where these features cannot be resolved.
Also, the HR04 sample does not use color information to estimate the M∗ of their sample, rather than use Jeans modeling.
This gives rise to systematic uncertainties related to the choiceof the initial mass function for example, which need to be kept
in mind. We refer the reader to Kormendy & Ho (2013) for
more discussion of systematic issues in local samples.
The issue of a consistent calibration of black hole masses
is perhaps the most subtle and thorny. Not all the local correlations are mutually consistent, especially when considering
different galaxy properties like luminosity, stellar mass, andvelocity dispersion, and therefore a single self-consistent calibration is not possible. To illustrate the problem, we compare
our local MBH -M∗ sample to that published by Kormendy &Ho (2013) (hereafter K13). We find that the MBH -M∗ measurements by K13 have a global ∼ +0.3 dex offset relative to
our local sample. Note that a +0.3 dex difference would almost erase the ∆ logMBH for the high redshift sample as described in the last section, if it were independent of the blackhole mass calibration. However, the MBH -σ∗ relations by
K13 are also offset by ∼ +0.3 dex with respect to those given
by Woo et al. (2010), which is the baseline for our adopted
black hole mass estimators. That is, if we were to adopt theK13 normalization instead, the MBH of all the AGN sample should be increased by ∼0.3 dex at the same time. As aresult, adopting a different local anchor would only globallyaffect the level of the MBH to local and high-z sample at thesame time, with the result of leaving ∆ logMBH unchanged.
To illustrate this point further, we adopt the K13 local ellipticals and classical bulges as the z ∼ 0 anchor and compare
to the recalibrated MBH of our high-z AGNs sample. The
inference of γ changes to 0.74 ± 0.31 which is consistent at
1σ level with the value (i.e., γ = 1.03 ± 0.25 ) based on ourchosen anchor.
Bentz & Manne-Nicholas (2018), given the superior resolution of their data in the local universe, decomposed their hostusing more components than bulge+disk (e.g., Bar(s), Barlens, Ring(s)). This approach is impossible at high-z given
current data, and therefore their sample should not be used
as local anchor, for consistency. Just for illustration, when
adopting the Bentz & Manne-Nicholas (2018) as the local anchor, we obtain γ = 2.12 ± 0.32 using the 32 AGNs, indicating still a positive evolution, although as expected the amountof evolution is different for the reasons given above.
In conclusion, whereas the uncertainty in the local anchor
does not affect our measurement of evolution, it is importantto keep in mind this issue in evolutionary studies and verify
that the measurements are as self-consistent as possible in thelocal and distant universe. In order to further reduce the uncertainty, one should perform a self-consistent measurementwith identical techniques and data quality both in the local
and distant universe. This effort is currently in progress andwhen completed should enable us to completely eliminate thissource of uncertainty (Bennert et al. 2011b; Harris et al. 2012;
Bennert et al. 2015).

13

7. CONCLUSIONS
We studied the evolution of the correlations between the supermassive black hole and their host galaxies using new measurements of 32 X-ray selected AGNs at 1.2 < z < 1.7.
Near-infrared spectroscopic observations with Subaru/FMOSof the Hα emission line are available that provide reliable BH
masses. To obtain the properties of the AGN host galaxies,
we performed an image decomposition using state-of-the-art
techniques on HST/WFC3 IR data to obtain high-resolution
imaging of the AGN and its host. This required us to collect
PSF-stars across all the fields to build a library of PSFs. Usingstate-of-the-art image modeling tool L ENSTRONOMY, we decomposed the AGN image in the 2D plane taking each PSF in
the library. We obtained the host properties (i.e., host flux, effective radius, Sérsic index) using a weighted arithmetic mean
based on the inference from the eight top-ranked PSFs. With
additional HST/ACS image in the optical, we identified theappropriate stellar populations and derived the rest-frame Rband luminosity (Lhost ) and stellar mass (M∗ ) of the host withthe corresponding mass-to-light ratio. We then determine themass relations at high-z and establish any signs of evolution
by combining our high-z measurements with local and intermediate redshift samples from the literature (Park et al. 2015;
Bennert et al. 2011b; Schramm & Silverman 2013; Cisternas
et al. 2011).
We find the average “observed” ratio between the mass ofa SMBH and either their total host luminosity or total stellar
mass is larger at z ∼ 1.5, by 0.41 dex (a factor of ∼ 2.6) and
0.43 dex (a factor of ∼ 2.7), respectively, as compared to thatin the local universe. However, taking into account uncertainties and the bias due to selection effect could bias, the offset
to the local universe is only marginally significant. Even withthe remaining uncertainties in the corrections for selection, weare confident that any evolution in the MBH /M∗ relation is at
the most a factor of two with the case of no evolution entirely
plausible.
Considering the bulge component (Bennert et al. 2011b;
Woo et al. 2008) separately, we find that our high-z AGN sample has significantly lower mass/luminosity bulges at fixedblack hole mass than in the local universe. Comparing totheir MBH , we find evolution with the ratio evolving as(1 + z)2.09±0.30 , which is significant even when accounting
for selection effects. We caution that the evolution of thebulge component is more uncertain than our measurement of
the total galaxy components, since it is based on an estimateof the B/T ratio from the measured Sérsic index based on amatched sample of inactive galaxies. Nevertheless, since thesignal is stronger than for the integrated quantities, the measurement is more significant even given larger uncertainties.
Thus, we can conclude that the BHs 8 − 10 Gyrs ago reside inbulges which are less massive/luminous than today (see also
Bennert et al. 2011b; Park et al. 2015), even though a proper
measure of the bulge mass at high-z may need to wait for the
next generation ground-based 20-30m class telescopes.
We summarize several limitations of this work. First, due
to finite resolution of HST, we model the host as a singleSérsic profile. The residuals do not show evidence of additional components, indicating that our choice is appropriate
given the noise and resolution of the data. However, we cannot exclude that more features would be revealed by higher
quality data.
Second, we adopt simple stellar population templates to derive the host luminosity and stellar mass for the sample, which

14

X. Ding et al.

introduces some uncertainty in the inference of the host properties. However, this source of uncertainty is subdominantwith respect to that associated with single epoch black holemass estimates (see Section 6.1 for detail).
Third, for a few AGN hosts, their inferred host effective radius hits the lower limit, and thus the inference of their Reff
should be considered as the upper limit. However, the inference of the host luminosity and stellar mass is still reliable inthese cases, as shown by our test using different lower limit
boundaries (Section 5.1).
Fourth and last, the choice of local anchor affects our inferred evolution at some level. We adopted as our baseline thelocal samples for which measurements have been made witha procedure that is most similar to the one we have applied
at high-z, but other choices are possible. We investigated theimpact of different choices for the local anchor in detail and
we found that even though the numerical values can change
somewhat, the inferred evolution is always positive (see Section 6.2).
Given that the M∗,total -MBH relation is closer to or consistent with the local relation, the inferred evolution of theM∗,bulge -MBH correlations is qualitatively consistent with ascenario where the assembly of the black hole predates that of
the bulge, which is processed by the transfer of stellar massfrom the disk via mergers or secular processes (Jahnke et al.
2009; Bennert et al. 2011a; Schramm & Silverman 2013). The
stellar mass needed to build the bulge appears to be present in
the disk component at high-z.
Of further interest, the scatter in the MBH -M∗ ratios at
high-z is similar to the local one. This was unexpected since
the measurements of the masses of both the black holes andtheir hosts should have more uncertainty than local estimates.
If true, this may indicate that cosmic averaging of initially unrelated states (Peng 2007) are not driving the relation between
SMBHs and their host galaxies, since then a larger scatter athigher redshifts would be expected. Thus, there is likely aconnection between the two yet to be fully understood; contenders include AGN feedback or links to common gas reservoirs.
Finally, the forthcoming launch of the James Webb Space
Telescope (JWST) and the first light of adaptive-optics assisted

extremely large telescopes may provide high-quality imaging data of AGNs at higher redshift (up to z ∼ 7) and thus
trace the evolution of correlations at the more distant universe.
Although, galaxies, as we know, are smaller at high-z thus
may remain challenging to detect under the glare of a luminous AGN. In the lower redshift Universe, wide-area surveys
with Subaru/HSC, LSST, and WFIRST offer much promise
to build samples for studying these mass ratios and dependencies on other factors (e.g., environment).

ACKNOWLEDGMENTS
Based in part on observations made with the NASA/ESA
Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract
NAS 5-26555. These observations are associated with programs #15115. Support for this work was provided by NASAthrough grant number HST-GO-15115 from the Space Telescope Science Institute, which is operated by AURA, Inc.,
under NASA contract NAS 5-26555.
The authors thank the anonymous referee for helpful suggestions and comments which improved this paper. We acknowledge the importance of the conference “Particle Astrophysics and Cosmology Including Fundamental InteraCtions (PACIFIC-2013)” in preparing for this program. We
fully appreciate input from Luis Ho, Renyue Cen, Hyewon
Suh, Takahiro Morishita and Peter Williams. XD, SB andTT acknowledge support by the Packard Foundation througha Packard Research fellowship to TT. TT acknowledges support by NSF through grant AST-1412315. VNB gratefully
acknowledges assistance from a NASA grant associated with
HST proposal GO 15215. JS is supported by JSPS KAKENHI
Grant Number JP18H01251 and the World Premier International Research Center Initiative (WPI), MEXT, Japan. A.S.
is supported by the EACOA fellowship.
This work has made use of L ENSTRONOMY (Birrer &Amara 2018), A STROPY (Astropy Collaboration et al. 2013),
PHOTUTILS (Bradley et al. 2016), TOPCAT (Taylor 2005),
M ATPLOTLIB (Hunter 2007) and standard Python libraries.

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16

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APPENDIX
A. AGN - HOST GALAXY DECOMPOSITION FOR THE REMAINING 31 AGNS
The AGNs decomposition for the other 31 objects of our sample, presented in the same way as Figure 2.

data - Point Source

normalized residual

6

10

15 20

30

data
model
QSO
host+1objs

CID1281

40
30

1"

0

10 4

1"
10 3

10 2

10 1

data

100

10 4

1"
10 3

10 2

10 1

model

100

10 4

1"
10 3

10 2

10 1

data - Point Source

100 6

4

2

0

2

4

6

0.1

0.2

0.5

2

normalized residual

1.0

2.0

15 20

30

arcsec
pixel

4

6

10

data
model
QSO
host

CID206

40
30

10

1"

0

1"

10 5 10 4 10 3 10 2 10 1 100

70

data

1"

10 5 10 4 10 3 10 2 10 1 100

model

1"

10 5 10 4 10 3 10 2 10 1 100

data - Point Source

6

4

2

0

2

4

6

0.1

0.2

2

normalized residual

0.5

1.0

arcsec
4

6

pixel

10

2.0

15 20

30

1"
10 5

50

0.25
0.00
0.25
0.50

1"
10 4

10 3

10 2

10 1

data

100 10 5

1"
10 4

10 3

10 2

10 1

model

100 10 5

1"
10 4

10 3

10 2

10 1

data - Point Source

100 6

4

30

data
model
QSO
host+1objs

30

60

28

(mag, pixel 2)

CID216

40

0

26

28

50

10

32
0.25
0.00
0.25
0.50

26

60

20

29

30

20

80

28

31

10

50

27

30

20

60

26

(mag, pixel 2)

model

pixel

4

(mag, pixel 2)

50

data

2

2

0

2

4

6

0.1

34

0.2

2

normalized residual

32

0.5

1.0

arcsec
4

pixel
6

10

0.25
0.00
0.25
0.50

2.0

15 20

30

data
model
QSO
host+1objs

CID237

40
30

24
25

(mag, pixel 2)

60

26
27
28
29

20
10

1"

0
10 4

10 3

1"
10 2

10 1

100

10 4

10 3

1"
10 2

10 1

100

10 4

10 3

30

1"
10 2

10 1

100

6

4

2

0

2

4

6

0.1

0.2

0.5

arcsec

1.0

2.0

0.25
0.00
0.25
0.50

Cosmic evolution of the MBH -host relation

model

data - Point Source2

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23

TABLE 1D ETAILS OF O BSERVATION
Object ID
(1)

z
(2)

WFC3/Filter
(3)

RA
(4)

DEC
(5)

Observing date
(6)

COSMOS-CID1174
COSMOS-CID1281
COSMOS-CID206
COSMOS-CID216
COSMOS-CID237
COSMOS-CID25512
COSMOS-CID3242
COSMOS-CID3570
COSMOS-CID452
COSMOS-CID454
COSMOS-CID50COSMOS-CID543
COSMOS-CID597
COSMOS-CID607
COSMOS-CID70
COSMOS-LID1273
COSMOS-LID1538
COSMOS-LID360
COSMOS-XID2138
COSMOS-XID2202
COSMOS-XID2396
CDFS-1
CDFS-229
CDFS-321
CDFS-724
ECDFS-358
SXDS-X1136
SXDS-X50
SXDS-X717
SXDS-X735
SXDS-X763
SXDS-X969

1.552
1.445
1.483
1.567
1.618
1.664
1.532
1.244
1.407
1.478
1.239
1.301
1.272
1.294
1.667
1.617
1.527
1.579
1.551
1.516
1.600
1.630
1.326
1.570
1.337
1.626
1.325
1.411
1.276
1.447
1.412
1.585

F140W
F140W
F140W
F140W
F140W
F140W
F140W
F125W
F125W
F140W
F125W
F125W
F125W
F125W
F140W
F140W
F140W
F140W
F140W
F140W
F140W
F140W
F125W
F140W
F125W
F140W
F125W
F125W
F125W
F140W
F125W
F140W

150.2789
150.4160
149.8371
149.7918
149.9916
150.1017
149.7113
149.6411
150.0045
149.8681
150.2080
150.4519
150.5262
150.6097
150.4051
150.0565
150.6215
150.1251149.7036
150.6530149.4779
52.899053.068053.048653.2870
53.0850
34.8925
34.0267
34.5400
34.558134.5849
34.75941.95952.52582.00881.87291.7243
1.84832.1452
2.10762.23712.33072.08332.14482.24492.32312.2701
1.62752.15882.86172.57811.99692.6425-27.8600-27.6580-27.6239-27.6940-28.0370-5.1498-5.0602-5.0334-4.8781-4.7864-5.4291

2017-10-26
2018-11-26
2017-10-23
2017-10-23
2018-06-03
2019-03-16
2017-10-26
2017-10-27
2017-10-25
2018-02-26
2017-10-23
2018-04-30
2018-11-25
2017-10-25
2017-10-27
2017-10-31
2018-05-01
2017-10-30
2017-11-01
2017-11-05
2017-11-12
2018-04-03
2018-04-04
2018-08-18
2018-04-05
2018-02-09
2018-01-29
2018-03-01
2018-07-02
2017-11-14
2018-07-03
2018-07-02

N OTE. — Column 1: Object field and ID. Column 2: Spectroscopic redshift. Column 3:
WFC3 filter. Note that the targets from the COSMOS field also have ACS imaging. Column
4 and 5: J2000 RA and DEC coordinates. Column 6: The observing start date. The total
exposure time of each target is 2394s.

B. AGN HOST GALAXY LUMINOSITY WITH A CORRECTION FOR PASSIVE EVOLUTION
In the passive evolution scenario, we expect the galaxy luminosity to fade over time. Thus, we transfer the Lhost for distantsamples at today so as to compare the MBH -Lhost relation to the local in the equivalent frame. We consider this scenario
following Ding et al. (2017b) by parametrizing the luminosity evolution with the functional form as dmagR ∼ d log(1 + z), i.e.,log(LR,0 ) = log(LR ) − 1.48 log(1 + z).

(B1)

This formalism is more accurate to fit a broad range redshift comparing to a single slope as dmag/dz. We refer the interested
readers to Ding et al. (2017b, section 5.4) for more details.
Having transferred the Lhost to today, we find that, as showing in Figure 14-(a), at fixed mass, the BH in the more distantuniverse tends to reside in less luminous hosts, which is consistent to the MBH -M∗ relation. Based on the 32 AGN systems,
we fit the offset as a function of redshift in form as Equation 9 and obtain γ = 1.07 ± 0.23, as showing in Figure 14-(b). We
further consider the selection effect using the previous approach as introduced in Section 5.5, and obtain γ = 0.5 ± 0.5 andγ = 0.6 ± 0.4, with flat and lognormal prior, respectively, as showing in Figure 14-(c), (d).
The inferred γ here could have large systematics given the following limitations. First, the passive evolution is based on asimplified correction; after all, we are not clear exactly how the host evolves to z = 0. Moreover, we only consider the evolutionof the host galaxy and assume the MBH does not change much.
C. THE MBH -M∗ COMPARISON SAMPLE
We use the self-consistent recipes to re-calibrate the measurements of our comparison sample. We list the values of all thecomparison MBH -M∗ sample in the Table 7.

24

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Local AGNs by Bennert+10
Intermediate redshift AGNs
This work

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7.0
6.5

log(MBH/107M )=0.50+0.95log(LR/1010L )

9.0

9.5

10.0

10.5

11.0

log(LR, host/L )

11.5

12.0

0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
0.5
1.0
1.5
2.00.0z 1

0.5

Local AGNs by Bennert+10
Intermediate redshift AGNs

0.1

0.2

1.5

2

2.5

This work

0.3

log(1+z)

0.4

0.5(a) MBH -Lhost relation, evolution-corrected(b) offset in log(MBH ) (VS. Lhost ) as a function of redshift(c) MBH -Lhost , flat prior(d) MBH -Lhost , lognormal prior

F IG . 14.— Same as previous figures but for MBH -Lhost relation, considering the passive evolution correction for host galaxy luminosity.

Cosmic evolution of the MBH -host relation

25

TABLE 2
AGN PROPERTIES
Hα emission line

Target ID

Hβ emission line

(1)

FWHM
(km s−1 )(2)log(LHα )(erg s−1 )(3)logMBH(M )(4)Eddington ratio(log(LBol /LEdd ))(5)

FWHM(5100)(km s−1 )(6)log(Lλ5100 )(erg s−1 )(7)

logMBH(M )(8)Eddington ratio(log(LBol /LEdd ))(9)

CID1174
CID1281
CID206
CID216
CID237
CID255
CID3242
CID3570
CID452
CID454
CID50
CID543
CID597
CID607
CID70
LID1273
LID1538
LID360
XID2138
XID2202
XID2396
CDFS-1
CDFS-229
CDFS-321
CDFS-724
ECDFS-358
SXDS-X1136
SXDS-X50
SXDS-X717
SXDS-X735
SXDS-X763
SXDS-X969

1906
1619
3334
2230
2112
1932
2543
1959
3458
2824
2340
2189
1656
3009
2480
3224
2941
2482
3186
2973
2271
2000
2190
2442
2541
2237
2760
1817
2931
2702
2961
2296

43.43
43.24
43.48
42.85
43.86
43.99
43.83
43.16
42.92
43.34
43.94
43.57
43.33
43.67
43.51
43.61
43.60
43.88
43.61
43.56
44.06
43.02
43.60
43.93
42.95
43.40
43.43
43.42
43.05
43.70
43.57
43.50

7.99
7.75
8.53
7.85
8.29
8.27
8.45
7.89
8.30
8.31
8.42
8.19
7.81
8.53
8.27
8.56
8.47
8.45
8.55
8.46
8.46
7.83
8.20
8.46
8.03
8.12
8.33
7.94
8.20
8.44
8.47
8.20

-0.47
-0.41
-0.97
-0.90
-0.36
-0.22
-0.55
-0.63
-1.26
-0.88
-0.42
-0.53
-0.39
-0.78
-0.68
-0.87
-0.79
-0.50
-0.86
-0.82
-0.33
-0.03
-0.60
-0.46
-1.15
-0.64
-0.81
-0.43
-1.05
-0.67
-0.81
-0.61

5898
···
···
···
···
3709
3775
···
3127
···
1939
···
···
4242
3982
···
···
2869
2945
···
2658
···
···
···
···
···
6761
···
···
3520
4509
1696

44.76
···
···
···
···
45.37
45.10
···
44.63
···
45.33
···
···
44.78
45.16
···
···
45.09
44.81
···
45.50
···
···
···
···
···
44.71
···
···
45.07
44.51
45.05

8.83
···
···
···
···
8.73
8.61
···
8.22
···
8.15
···
···
8.56
8.69
···
···
8.37
8.25
···
8.51
···
···
···
···
···
8.93
···
···
8.54
8.47
7.90

-1.34
···
···
···
···
-0.60
-0.75
···
-0.88
···
-0.06
···
···
-1.04
-0.77
···
···
-0.52
-0.71
···
-0.24
···
···
···
···
···
-1.49
···
···
-0.70
-1.29
-0.08

N OTE. — Column 1: Object ID. Column 2-5: Hα emission line width (FWHM), Hα luminosity, inferred MBH and Eddington ratio. Column
6-9: Hβ emission line BH properties. The typical uncertainty level for FWHM is 15%, and for log(Lλ ) is in the range 0.01 ∼ 0.2 dex, respectively.
The inferred uncertainty level for MBH are assumed as 0.4 dex.

TABLE 3
H OST GALAXY MEASUREMENTS OF CID1174
PSF rank(1)

total χ2
(2)

weights wi(3)

host flux (counts)(4)

host flux ratio(5)

Reff (arcsec)(6)Sérsic n
(7)

1
2
3
4
5
6
7
8

8584.429
8646.711
8816.947
9304.841
9652.575
9917.101
10018.324
10087.456

1.000
0.920
0.734
0.383
0.241
0.170
0.148
0.135

82.2
99.1
76.7
128.6
187.5
100.2
75.1
79.8

35%
42%
33%
55%
79%
42%
32%
34%

0.00 345
0.00 298
0.00 365
0.00 231
0.00 116
0.00 287
0.00 365
0.00 358

1.1
1.9
1.1
2.8
6.2
2.1
1.2
1.2

97.322 ± 28.336

42% ± 12%0.00 309 ± 0.00 065

1.9 ± 1.3

Weighted value

N OTE. — Column 1: Rank of the PSF from the library. Column 2: Total χ2 for the corresponding PSF. Column 3:
Weights for the inference. Column 4-7: Fitted value for the host flux, host/total flux ratio, effective radius, and Sérsic
index. For this sample, the inflation parameter α calculated by Equation 4 is 16.671.

26

X. Ding et al.
TABLE 4
H OST GALAXY PROPERTIES

Target ID(1)
CID1174
CID1281

WFC3
χ2
host-total flux ratio(reduced)
(2)
(3)
2.307
1.322

ACS/F814W

Reff
(00 )
(4)

magnitudeχ2
host-total flux ratio magnitude(AB system) (reduced)(AB system)
(6)
(7)
(8)
(9)

Sérsic n
(5)

+0.37
21.48−0.28

2.496

11% ± 1%

+0.11
+0.15
+0.18
+0.40
23.21−0.10
11.01−0.11
10.63−0.15
10.21−0.53

49% ± 14%+0.36
22.88−0.27+1.30
21.82−0.58+0.05
21.51−0.05+0.26
21.28−0.21+0.37
21.61−0.28+0.35
21.16−0.26+0.02
21.16−0.02+0.06
21.18−0.06+0.08
21.20−0.07+0.86
20.93−0.48+0.41
21.99−0.30+0.54
21.87−0.36+0.58
21.19−0.37+0.30
21.86−0.24+0.21
20.94−0.18+0.22
21.25−0.18+0.14
21.46−0.12+0.17
21.87−0.15+0.30
21.16−0.24+0.65
21.40−0.40+0.40
22.47−0.29+0.14
21.57−0.13+0.70
20.34−0.42+0.58
23.70−0.38+0.30
21.34−0.24+0.23
21.92−0.19+0.27
21.99−0.21+0.18
21.76−0.15+0.33
20.92−0.25+1.17
24.13−0.55+0.52
21.59−0.35

1.378

19% ± 8%

+0.60
+0.15
+0.18
24.83−0.38
10.38−0.11
10.00−0.15+0.28
9.77−0.51

1.903

8% ± 2%+0.52
10.14−0.62

1.425

35% ± 2%2.354

3% ± 2%2.858

4% ± 2%2.596

5% ± 1%1.332

86% ± 2%1.452

38% ± 1%1.291

9% ± 1%4.940

5% ± 3%1.435

5% ± 2%1.254

12% ± 1%2.590

5% ± 2%2.361

2% ± 1%

2.137

6% ± 1%2.173

8% ± 1%4.914

4% ± 1%2.731

5% ± 1%3.852

8% ± 2%5.346

2% ± 1%+0.40
23.67−0.29+0.05
23.45−0.05+1.04
23.72−0.52+0.60
22.89−0.39+0.34
23.60−0.26+0.01
22.97−0.01+0.02
22.73−0.02+0.06
23.35−0.06+1.15
22.50−0.55+0.53
23.77−0.36+0.13
23.56−0.11+0.51
23.57−0.35+0.68
24.63−0.41+0.15
23.29−0.13+0.19
23.09−0.16+0.17
23.25−0.15+0.31
23.90−0.24+0.29
22.59−0.23+0.24
23.36−0.20

···

···

···

···

···

···

···

···

···

···

···

···

···

···

···

···

···

···

···

···

···

···

···

···

···

···

···

···

···

···

···

···

···

35% ± 24%

CID2161.514

94% ± 5%0.00 25 ± 0.00 06 6.2 ± 1.2

CID2372.349

30% ± 6%0.00 87 ± 0.00 17 4.7 ± 1.7

19% ± 5%0.00 19 ± 0.00 06 4.2 ± 1.5

46% ± 13%0.00 20 ± 0.00 16 6.1 ± 1.9

77% ± 2%0.00 70 ± 0.00 01 0.7 ± 0.1

75% ± 4%0.00 37 ± 0.00 02 1.4 ± 0.2

CID3570
CID452

2.751
1.665
1.684

CID4542.203

36% ± 3%0.00 39 ± 0.00 02 0.6 ± 0.1

CID50

5.576

17% ± 9%0.00 16 ± 0.00 11 3.2 ± 2.2

CID5431.902

31% ± 10%0.00 10 ± 0.00 00 0.5 ± 0.3

42% ± 17%0.00 17 ± 0.00 06 1.8 ± 0.8

44% ± 18%0.00 21 ± 0.00 09 3.4 ± 1.1

20% ± 5%0.00 42 ± 0.00 10 3.6 ± 1.0

53% ± 9%0.00 30 ± 0.00 04 1.2 ± 0.5

CID597
CID607
CID70
LID1273

1.565
1.692
2.041
1.697

LID15382.362

44% ± 8%0.00 18 ± 0.00 04 2.8 ± 0.5

LID3603.918

18% ± 2%0.00 63 ± 0.00 02 0.8 ± 0.4

XID21381.597

39% ± 6%0.00 50 ± 0.00 03 1.2 ± 0.4

33% ± 8%0.00 10 ± 0.00 00 4.0 ± 1.0

24% ± 11%0.00 58 ± 0.00 09 0.8 ± 1.4

65% ± 20%0.00 14 ± 0.00 07 4.8 ± 1.1

18% ± 2%0.00 51 ± 0.00 03 0.5 ± 0.2

XID2202
XID2396
CDFS-1
CDFS-229

3.23
3.669
1.358
4.329

CDFS-321

3.998

25% ± 12%0.00 38 ± 0.00 12 2.3 ± 2.0

CDFS-724

1.355

35% ± 15%0.00 12 ± 0.00 03 1.6 ± 1.1

56% ± 14%0.00 36 ± 0.00 04 1.7 ± 0.5

41% ± 8%0.00 10 ± 0.00 00 2.0 ± 0.5

41% ± 9%0.00 19 ± 0.00 04 1.7 ± 0.6

61% ± 9%0.00 26 ± 0.00 07 5.6 ± 1.4

ECDFS-358
SXDS-X1136
SXDS-X50
SXDS-X717

2.012
1.937
1.423
1.426

log M∗,bulge
(M )
(12)

0.00 24 ± 0.00 09 3.2 ± 1.5

2.054

CID3242log M∗(M )(11)42% ± 12%

CID2061.625

log LR(L ,R )
(10)

0.00 31 ± 0.00 07 1.9 ± 1.3
0.00 29 ± 0.00 15 3.1 ± 2.5

CID255Derived properties

SXDS-X735

2.203

32% ± 9%0.00 22 ± 0.00 06 2.0 ± 1.0

SXDS-X763

2.376

6% ± 4%0.00 69 ± 0.00 53 2.4 ± 0.8

SXDS-X969

1.613

29% ± 11%0.00 11 ± 0.00 02 2.1 ± 1.1

+0.52
+0.53
10.83−0.23
10.45−0.25
+0.03
+0.10
11.01−0.03 10.63−0.10
+0.11
+0.14
11.13−0.09
10.75−0.13
+0.18
+0.15
11.03−0.11 10.65−0.15
+0.14
+0.17
11.12−0.11 10.75−0.15
+0.02
+0.10
10.98−0.02 10.71−0.10
+0.03
+0.10
11.13−0.03 10.86−0.10
+0.04
+0.10
11.07−0.04 10.70−0.10
+0.35
+0.36
11.07−0.19
10.80−0.21
+0.19
+0.16
10.43−0.15
10.70−0.12
+0.24
+0.22
10.73−0.15 10.46−0.18
+0.25
+0.23
11.02−0.15 10.75−0.18
+0.12
+0.16
10.93−0.10 10.55−0.14
+0.09
+0.13
11.27−0.07
10.89−0.12
+0.09
+0.13
11.09−0.08 10.71−0.12
+0.06
+0.11
11.04−0.05 10.66−0.11
+0.07
+0.12
10.85−0.06
10.48−0.12
+0.12
+0.16
11.11−0.10 10.74−0.14
+0.28
+0.26
11.07−0.16 10.69−0.19
+0.19
+0.16
10.29−0.15
10.66−0.12
+0.06
+0.12
10.90−0.05 10.63−0.11
+0.28
+0.30
11.48−0.17
11.10−0.20
+0.23
+0.25
10.06−0.15
9.79−0.18
+0.12
+0.16
11.11−0.10 10.73−0.14
+0.09
+0.14
10.75−0.08 10.49−0.13
+0.15
+0.11
10.54−0.13
10.80−0.09
+0.12
+0.07
10.77−0.06 10.51−0.12
+0.17
+0.13
10.78−0.14
11.16−0.10
+0.47
+0.48
9.95−0.22 9.68−0.24
+0.21
+0.23
10.99−0.14
10.61−0.17+0.15
10.52−0.25+0.20
10.60−0.34+0.23
10.48−0.35+0.19
10.62−0.29+0.68
9.78−0.13+0.48
10.13−0.29+0.74
9.77−0.15+0.41
10.51−0.56+0.64
9.55−0.24+0.44
9.98−0.49+0.29
10.57−0.41+0.22
10.38−0.34+0.58
10.13−0.28+0.22
10.52−0.37+0.65
9.76−0.16+0.57
9.71−0.28+0.21
10.58−0.33+0.58
10.00−0.40+0.22
10.14−0.31+0.70
9.71−0.16+0.42
10.72−0.58+0.48
9.25−0.48+0.44
10.22−0.47+0.36
10.10−0.48+0.45
10.01−0.46+0.18
10.38−0.29+0.36
10.42−0.56+0.48
9.35−0.56+0.40
10.21−0.54

N OTE. — Column 1: Object ID. Column 2-6: WFC3 inference. Column 2: Reduced χ2 value by the best PSF in the library. Column 7-9: ACS inference.
Column 10: Observed host luminosity in the rest-frame R band. Column 11: Host total stellar mass. Column 12: Bulge stellar mass, using Sérsic index as B/T proxy,see Section 5.6 for the details.

TABLE 5
S UMMARY OF γ INFERENCES

SampleSelection effects MBH -M∗ MBH -Lhost

32 AGNs + intermediateNo0.72±0.20 0.64±0.17
32 AGNs + intermediate
Yes
0.90±0.40 0.50±0.50
32 AGNs
No
1.03±0.25 1.07±0.23
32 AGNs
Yes
1.50±0.80 1.80±0.80
N OTE. — Entire sample includes the 32 AGNs and the intermediate redshift AGNs from the reference in Section 2.4. Note that the adopted Lhost have beentransferred to today assuming the passive evolution scenario using Equation B1. The results of selection effects used the uniform (flat) prior of σint .

Cosmic evolution of the MBH -host relation
TABLE 6O BSERVED γ FOR MBH -M∗ WITHOUT SELECTION EFFECT CORRECTIONS

Sample combinationγ

32 AGNs
1.03±0.25
30 AGNs (excluded outliers) 0.89±0.27
32 AGNs + intermediate 0.72±0.20
32 AGNs bulge only
2.09±0.30
N OTE. — These values are directly inferred for the observation, without corrections due to selection effects.

27

28

X. Ding et al.
TABLE 7DATA SUMMARY FOR THE COMPARISON MBH -M∗ SAMPLE

Object IDa

z

logM∗
(M )

Local sample by Häring & Rix (2004)
M87
0.0037b11.78c ±0.18
NGC1068
0.003510.36±0.18
NGC3379
0.002510.83±0.18
NGC4374
0.0043
11.56±0.18
NGC4261
0.0073
11.56±0.18
NGC6251
0.024311.75±0.18
NGC7052
0.0136
11.46±0.18
NGC47420.00369.79±0.18
NGC8210.0056
11.11±0.18
IC14590.0068
11.46±0.18
M310.0002
10.57±0.18
M320.00028.90±0.18
NGC1023
0.0027
10.84±0.18
NGC27780.005310.88±0.18
NGC3115
0.002311.08±0.18
NGC32450.004910.83±0.18
NGC3377
0.002610.20±0.18
NGC3384
0.002710.30±0.18
NGC36080.005310.99±0.18
NGC4291
0.0061
11.11±0.18
NGC43420.003610.08±0.18
NGC4473
0.003710.96±0.18
NGC45640.003510.64±0.18
NGC4594
0.002311.43±0.18
NGC4649
0.003911.69±0.18
NGC4697
0.002711.04±0.18
NGC58450.006010.57±0.18
NGC7332
0.005310.18±0.18
NGC7457
0.00319.85±0.18
Milky Way
0.000010.04±0.18
Local sample by Bennert et al. (2011a)
0121−0102
0.054010.12±0.240206−0017
0.043010.95±0.230353−0623
0.076010.33±0.220802+3104
0.041010.38±0.230846+2522
0.051010.50±0.231042+0414
0.052410.32±0.231043+1105
0.04759.83±0.24
1049+2451
0.055010.41±0.231101+1102
0.035510.33±0.221116+4123
0.021010.20±0.221144+3653
0.038010.26±0.221210+3820
0.02299.94±0.24
1250−0249
0.047010.14±0.221323+2701
0.05599.65±0.22
1355+3834
0.0501
10.11±0.231405−0259
0.054110.04±0.231419+0754
0.055810.73±0.211434+4839
0.036510.30±0.241535+5754
0.030410.24±0.241545+1709
0.04819.92±0.22
1554+3238
0.048310.00±0.231557+0830
0.04659.82±0.23
1605+3305
0.05329.95±0.23
1606+3324
0.058510.33±0.221611+5211
0.040910.33±0.22

logMBH(M )9.48±0.2
7.15±0.1
8.00±0.1
8.63±0.2
8.72±0.3
8.72±0.3
8.52±0.2
7.15±0.2
7.57±0.2
9.40±0.4
7.65±0.2
6.40±0.3
7.64±0.57.15±0.2
9.00±0.2
8.32±0.3
8.00±0.6
7.20±0.4
8.28±0.3
8.49±0.2
8.48±0.2
8.04±0.6
7.75±0.4
9.00±0.3
9.30±0.0
8.23±0.68.38±0.1
7.11±0.2
6.54±0.1
6.57±0.2
7.44±0.4
8.01±0.4
7.44±0.4
7.17±0.4
8.52±0.4
6.98±0.4
7.67±0.4
7.84±0.4
7.77±0.4
7.37±0.4
7.36±0.4
7.50±0.4
7.65±0.4
6.98±0.4
7.96±0.4
7.08±0.4
7.51±0.4
7.48±0.4
7.83±0.4
7.28±0.4
7.63±0.4
7.55±0.4
7.78±0.4
7.22±0.4
7.34±0.4

Object ID

z

logM∗
(M )

Intermediate sample by Bennert et al. (2011b)
J033252−275119
1.227
10.58(9.83)d ±0.2
J033243−274914
1.900
10.64(10.64)±0.2
J033239−274601
1.220
10.54(10.54)±0.2
J033226−274035
1.031
10.78(9.53)±0.2
J033225−274218
1.617
10.61(10.61)±0.2
J033210−274414
1.615
10.45(10.45)±0.2
J033200−274319
1.037
9.62(9.62)±0.2
J033229−274529
1.218
10.71(10.71)±0.2
J123553+621037
1.371
10.90(9.99)±0.2
J123618+621115
1.021
10.96(9.29)±0.2
J123707+622147
1.450
10.74(10.74)±0.2
Intermediate sample by Schramm & Silverman (2013)
158
0.717
10.66(10.66)±0.2
170
1.065
9.80(9.50)±0.2
271
0.960
10.49(9.85)±0.2
273
0.970
10.29(10.29)±0.2
305
0.544
11.01(11.01)±0.2
333
1.044
10.33(9.87)±0.2
339
0.675
10.96(10.96)±0.2
348
0.569
10.55(10.15)±0.2
379
0.737
10.66(10.17)±0.2
413
0.664
10.19(9.80)±0.2
417
0.837
10.39(9.91)±0.2
465
0.740
10.76(10.14)±0.2
516
0.733
10.88(10.71)±0.2
540
0.622
10.88(10.57)±0.2
597
1.034
10.91(10.80)±0.2
712
0.841
11.34(11.19)±0.2
Intermediate sample by Cisternas et al. (2011)
J095817.54+021938.5
0.73
10.30±0.2
J095819.88+022903.6
0.34
11.23±0.2
J095831.65+024901.6
0.34
10.65±0.2
J095840.61+020426.6
0.34
11.02±0.2
J095845.80+024634.0
0.35
10.54±0.2
J095902.76+021906.5
0.34
11.14±0.2
J095909.53+021916.5
0.38
10.68±0.2
J095928.31+022106.9
0.35
10.95±0.2
J100002.21+021631.8
0.85
11.07±0.2
J100012.91+023522.8
0.70
11.17±0.2
J100014.55+023852.7
0.44
10.57±0.2
J100017.54+020012.6
0.35
10.47±0.2
J100025.25+015852.2
0.37
10.57±0.2
J100028.63+025112.7
0.77
10.86±0.2
J100029.69+022129.7
0.73
11.01±0.2
J100033.38+015237.2
0.83
10.81±0.2
J100033.49+013811.6
0.52
10.54±0.2
J100037.29+024950.6
0.73
10.36±0.2
J100043.15+020637.2
0.36
11.28±0.2
J100046.72+020404.5
0.55
11.08±0.2
J100058.71+022556.2
0.69
10.66±0.2
J100118.52+015543.0
0.53
10.84±0.2
J100141.09+021300.0
0.62
10.53±0.2
J100146.49+020256.7
0.67
10.75±0.2
J100202.22+024157.8
0.79
10.53±0.2
J100205.03+023731.5
0.52
11.15±0.2
J100212.11+014232.4
0.37
10.48±0.2
J100218.32+021053.1
0.55
11.20±0.2
J100230.06+014810.4
0.63
10.73±0.2
J100230.65+024427.6
0.82
10.77±0.2
J100232.13+023537.3
0.66
11.03±0.2
J100243.96+023428.6
0.38
11.08±0.2

N OTE. —a : The object name is directly adopted from the reference.
b : For the Häring & Rix (2004) sample, the redshift value is calculated from the Galaxy distance, assuming perfect Hubble flow.
c : M for the local sample are the bulge masses.
∗d : If applicable, the values in brackets are the bulge mass.

logMBH(M )8.87±0.4
9.17±0.4
8.24±0.4
7.85±0.4
8.08±0.48.30±0.4
7.75±0.4
8.37±0.4
8.27±0.4
8.35±0.4
8.77±0.4
6.94±0.4
6.84±0.4
7.12±0.4
7.99±0.4
8.38±0.4
7.66±0.4
7.60±0.4
7.64±0.4
8.90±0.4
6.81±0.4
8.11±0.4
7.78±0.4
7.66±0.4
7.38±0.4
7.87±0.4
8.41±0.4
7.72±0.4
8.29±0.4
8.08±0.4
8.39±0.4
7.39±0.4
8.66±0.4
7.77±0.4
7.24±0.4
8.29±0.4
8.15±0.4
7.79±0.4
7.59±0.4
8.58±0.4
8.49±0.4
8.03±0.4
8.07±0.4
8.01±0.4
7.41±0.4
8.07±0.4
7.75±0.4
7.91±0.4
8.22±0.4
7.35±0.4
7.73±0.4
8.24±0.4
8.38±0.4
7.70±0.4
8.61±0.4
7.50±0.4
7.82±0.4
8.19±0.4
8.25±0.4