A Molecular Einstein Ring at z=4.12: Imaging the Dynamics of a Quasar Host Galaxy Through a Cosmic Lens

Abstract

We present high-resolution (0.3") Very Large Array (VLA) imaging of the molecular gas in the host galaxy of the high redshift quasar PSS J2322+1944 (z=4.12). These observations confirm that the molecular gas (CO) in the host galaxy of this quasar is lensed into a full Einstein ring, and reveal the internal dynamics of the molecular gas in this system. The ring has a diameter of ~1.5", and thus is sampled over ~20 resolution elements by our observations. Through a model-based lens inversion, we recover the velocity gradient of the molecular reservoir in the quasar host galaxy of PSS J2322+1944. The Einstein ring lens configuration enables us to zoom in on the emission and to resolve scales down to ~1 kpc. From the model-reconstructed source, we find that the molecular gas is distributed on a scale of 5 kpc, and has a total mass of M(H2)=1.7 x 10^10 M_sun. A basic estimate of the dynamical mass gives M_dyn = 4.4 x 10^10 (sin i)^-2 M_sun, that is, only ~2.5 times the molecular gas mass, and ~30 times the black hole mass (assuming that the dynamical structure is highly inclined). The lens configuration also allows us to tie the optical emission to the molecular gas emission, which suggests that the active galactic nucleus (AGN) does reside within, but not close to the center of the molecular reservoir. Together with the (at least partially) disturbed structure of the CO, this suggests that the system is interacting. Such an interaction, possibly caused by a major `wet' merger, may be responsible for both feeding the quasar and fueling the massive starburst of 680 M_sun/yr in this system, in agreement with recently suggested scenarios of quasar activity and galaxy assembly in the early universe. Comment: 9 pages, 7 figures, to appear in ApJ (accepted June 27, 2008)

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arXiv:0806.4616v1 [astro-ph] 27 Jun 2008
draft version June 27, 2008, Accepted for Publication in the Astrophysical Journal
Preprint typeset using LATEX style emulateapj v. 04/21/07
A MOLECULAR EINSTEIN RING AT Z=4.12:
IMAGING THE DYNAMICS OF A QUASAR HOST GALAXY THROUGH A COSMIC LENS
Dominik A. Riechers1,2,7, Fabian Walter1, Brendon J. Brewer3, Christopher L. Carilli4,
Geraint F. Lewis3, Frank Bertoldi5, and Pierre Cox6
draft version June 27, 2008, Accepted for Publication in the Astrophysical Journal
ABSTRACT
We present high-resolution (0.3′′) Very Large Array (VLA) imaging of the molecular gas in the host
galaxy of the high redshift quasar PSSJ2322+1944 (z = 4.12). These observations confirm that the
molecular gas (CO) in the host galaxy of this quasar is lensed into a full Einstein ring, and reveal the
internal dynamics of the molecular gas in this system. The ring has a diameter of ∼1.5′′, and thus is
sampled over ∼20 resolution elements by our observations. Through a model-based lens inversion, we
recover the velocity gradient of the molecular reservoir in the quasar host galaxy of PSSJ2322+1944.
The Einstein ring lens configuration enables us to zoom in on the emission and to resolve scales down
to ?1kpc. From the model-reconstructed source, we find that the molecular gas is distributed on a
scale of 5kpc, and has a total mass of M(H2) = 1.7×1010M⊙. A basic estimate of the dynamical mass
gives Mdyn= 4.4 × 1010sin−2iM⊙, that is, only ∼2.5 times the molecular gas mass, and ∼30 times
the black hole mass (assuming that the dynamical structure is highly inclined). The lens configuration
also allows us to tie the optical emission to the molecular gas emission, which suggests that the active
galactic nucleus (AGN) does reside within, but not close to the center of the molecular reservoir.
Together with the (at least partially) disturbed structure of the CO, this suggests that the system
is interacting. Such an interaction, possibly caused by a major ‘wet’ merger, may be responsible for
both feeding the quasar and fueling the massive starburst of 680M⊙yr−1in this system, in agreement
with recently suggested scenarios of quasar activity and galaxy assembly in the early universe.
Subject headings: galaxies: active, starburst, formation, high redshift — cosmology: observations —
radio lines: galaxies
1. INTRODUCTION
A fundamental aspect in studies of galaxy formation
and evolution is to understand the connection between
AGN and starburst activity. The existence of a physical
connection between both processes is suggested by the
finding that present day galaxies show a strong relation-
ship between the mass of their central supermassive black
holes (SMBHs) and the mass and concentration of their
stellar spheroids (Magorrian etal. 1998; Ferrarese & Mer-
ritt 2000; Gebhardt etal. 2000; Graham etal. 2001). If
these relations were due to a coevolution of both compo-
nents during the early assembly of a galaxy, high-redshift
quasars and their associated host galaxies would be ideal
objects to study the active formation of both SMBHs
and bulge stars.
Studies of molecular gas (most commonly rotational
transitions of CO), the prerequisite material that fuels
star formation, have become an important tool to probe
Electronic address: dr@caltech.edu
1Max-Planck-Institut f¨ ur Astronomie, K¨ onigstuhl 17, Heidel-
berg, D-69117, Germany
2Astronomy Department, California Institute of Technology,
MC 105-24, 1200 East California Boulevard, Pasadena, CA 91125
3Institute of Astronomy, School of Physics, A28, University of
Sydney, NSW 2006, Australia
4National Radio Astronomy Observatory, PO Box O, Socorro,
NM 87801
5Argelander-Institut f¨ ur Astronomie, Universit¨ at Bonn, Auf
dem H¨ ugel 71, Bonn, D-53121, Germany
6Institut de RadioAstronomie Millim´ etrique, 300 Rue de la
Piscine, Domaine Universitaire, F-38406 Saint Martin d’H` eres,
France
7Hubble Fellow
distant quasar host galaxies, and revealed large molec-
ular gas reservoirs of >1010M⊙ in a number of these
sources (see Solomon & Vanden Bout 2005 for a gen-
eral review).These galaxies typically show huge far-
infrared (FIR) luminosities in excess of 1013L⊙, which
are thought to be powered by starbursts (and possibly a
central AGN; e.g., Omont et al. 2001; Wang et al. 2008).
Observations of molecular gas trace the regions that can
host massive starbursts. In addition, the velocity struc-
ture of molecular line emission has the potential to con-
strain the dynamical state of galaxies out to the earliest
epochs.
Rotational molecular line emission typically emerges
at FIR to radio wavelengths, i.e., in the limited wave-
length regime where the AGN in distant quasars does
not outshine all other emission. However, the cosmologi-
cal distances of high redshift quasars make it difficult to
resolve the faint emission from their host galaxies a such
long wavelengths. The physical resolution of such obser-
vations is in some cases boosted by gravitational lenses
acting as natural telescopes. The gravitational lensing
effect also magnifies the observed flux of the background
galaxy, in particular for systems in Einstein ring config-
urations. Due to the compactness of the AGN, optical
quasars in Einstein ring lens configurations are rare. Due
to their greater extent, the host galaxies of quasars are
much more likely to cross the inner Einstein ring caustic
of a gravitational lens.
In this paper, we report on high (0.3′′) angular reso-
lution Very Large Array (VLA)8observations of CO in
8The Very Large Array is a facility of the National Radio As-

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2Riechers et al.
the host galaxy of the z=4.12 quasar PSSJ2322+1944,
one of only two known z>4 galaxies that are both grav-
itationally lensed and detected in molecular gas emis-
sion (the other being BRI0952–0115 at z=4.43; Guil-
loteau etal. 1999). This galaxy was identified in a spec-
troscopic follow-up study of the Palomar Sky Survey
(DPOSS; Djorgovski et al. 2000), and found to be a
strongly lensed optical quasar (S. G. Djorgovski, private
communication). It was subsequently detected in hard
X-ray (Vignali et al. 2005), FIR dust (Omont et al. 2001;
Isaak et al. 2002) and radio continuum emission (Carilli
et al. 2001), as well as molecular line emission (Cox et
al. 2002; Carilli et al. 2002). It follows the radio-FIR
correlation of star-forming galaxies (Carilli et al. 2001;
Beelen et al. 2006), indicating that its FIR continuum
emission is dominated by intense star formation. In spite
of the fact that this source shows only two unresolved
quasar images in the optical, previous CO observations
have shown that the molecular gas reservoir in its host
galaxy is lensed into an Einstein ring (Carilli etal. 2003;
hereafter: C03). These observations were also used to
derive a first lensing model for this source. Based on the
dynamical structure revealed by our new, higher resolu-
tion observations of PSSJ2322+1944, we have developed
a new lensing model, which enables us to reconstruct the
velocity gradient in the spatially resolved gas reservoir.
We use a concordance, flat ΛCDM cosmology through-
out, with H0=71 kms−1Mpc−1, ΩM=0.27, and ΩΛ=0.73
(Spergel etal. 2003, 2007).
2. OBSERVATIONS
We observed the CO(J=2→1) transition (νrest =
230.53799GHz) towards PSSJ2322+1944 using the VLA
in B configuration between 2006 June 19 and July
3, and in C configuration between 2002 October 21
and November 15 (these short spacings data were pub-
lished in the original study by C03).
sky integration time in the 11 observing runs amounts
to 70.5hr. At z = 4.119, the line is redshifted to
45.0351GHz (6.66mm). Observations were performed in
fast-switching mode (see, e.g., Carilli & Holdaway 1999)
using the nearby source J23307+11003for secondary am-
plitude and phase calibration. Observations were carried
out under very good weather conditions with 26 or 27
antennas. The phase stability in all runs was excellent
(typically <20◦phase rms for the longest baselines). The
phase coherence was checked by imaging the a calibra-
tor (J23207+05138) with the same calibration cycle as
that used for the target source. For primary flux cal-
ibration, 3C48 was observed during each run. Due to
the restrictions of the VLA correlator, one 50MHz in-
termediate frequency (IF) with seven 6.25MHz channels
was centered at the CO(J=2→1) line frequency, leading
to an effective bandwidth of 43.75MHz (corresponding
to 291kms−1at 45.0GHz). This encompasses almost
the full CO line width as measured in the CO(J=5→4)
transition (∆vFWHM= 273±50 kms−1, Cox etal. 2002).
Earlier observations set a 2σ limit of 150µJy on the con-
tinuum emission at the line frequency (Carilli etal. 2002).
For data reduction and analysis, the AIPS package was
used. All data were mapped using the CLEAN algorithm
The total on-
tronomy Observatory, operated by Associated Universities, Inc.,
under a cooperative agreement with the National Science Founda-
tion.
DECLINATION (J2000)
RIGHT ASCENSION (J2000)
23 22 07.2607.2207.1807.1407.10
19 44 23.0
22.5
22.0
21.5
21.0
Fig.1.— VLA map of the CO(J=2→1) emission to-
ward PSSJ2322+1944 (integrated over the central 37.5MHz, or
250kms−1).Contours are shown at (-3, -2, 1, 2, 3, 4, 5)×σ
(1σ = 48µJy beam−1). The beam size (0.33′′×0.30′′) is shown
in the bottom left corner. The large crosses show the positions of
the quasar images at λobs= 1.6µm (λr = 314nm), and the small
cross shows the position of the lensing galaxy at the same λ.
and natural weighting. The synthesized clean beam us-
ing all data has a size of 0.33′′×0.30′′.9The final rms
over a bandwidth of 37.5MHz (250 kms−1, excluding the
noisy edge channel) is 48µJy beam−1. In addition, seven
velocity channel maps (6.25MHz, or 42 kms−1each) of
the CO(J=2→1) emission were created. The rms after
Hanning smoothing is 77µJybeam−1. Convolving the
data to a linear spatial resolution of 0.5′′leads to slightly
higher rms values of 49 and 80µJy beam−1for maps at
250 and 42 kms−1velocity resolution.
3. RESULTS
In Figure 1, the velocity-integrated CO(J=2→1) emis-
sion over the central 250 kms−1is shown. The emission
is clearly resolved over multiple beams, and extended on
a scale of ∼1.5′′. The distribution of the gas is remi-
niscent of a full, almost circular molecular Einstein ring,
consistent with previous indications of such a structure in
lower resolution observations (C03). The emission varies
in intensity along the ring, showing clear substructure.
The apparent surface brightness variations along the
ring set strong constraints on the geometry of the lens
configuration. The brightest CO peak on the ring can
be used to set a lower limit on the intrinsic brightness
temperature of the molecular gas. The peak strength
of 280±48µJybeam−1corresponds to a beam-averaged,
rest-frame brightness temperature of Tb=8.7±1.5K. This
is by a factor of a few lower than the kinetic gas temper-
ature Tkinas predicted from CO line excitation models
of this source (Riechers et al. 2006; A. Weiß et al., in
prep.), and indicates that the substructure of the molec-
ular reservoir is not fully resolved by the observations.
We derive a spatially integrated CO(J=2→1) line peak
flux density of 2.50 ± 0.32mJy, which is consistent with
that found by Carilli etal. (2002). Assuming constant Tb
9Imaging B array data only gives a resolution of 0.17′′×0.15′′
(natural weighting), or 0.12′′×0.11′′(uniform weighting); however,
most of the emission is outresolved in such maps.

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High–Resolution CO(2–1) Imaging of PSSJ2322+19443
between CO(J=2→1) and CO(J=1→0), and a CO lu-
minosity to H2mass conversion factor for ultraluminous
infrared galaxies [α=0.8M⊙(K kms−1pc2)−1; Downes
& Solomon 1998], this corresponds to a total molecular
gas mass of M(H2) = 9.0 × 1010µ−1
the lensing magnification factor, see below). Within the
uncertainties, this is in agreement with the value found
by Riechers etal. (2006) based on the CO(J=1→0) lu-
minosity.
To increase the peak signal–to–noise ratio, Figure 2
shows the CO(J=2→1) emission convolved to 0.5′′lin-
ear resolution. This contour map is shown overlaid on
a greyscale image of the (rest-frame) 314nm contin-
uum emission of the source, as observed by the Hub-
ble Space Telescope (HST/NICMOS-2 F160W; image
adopted from Peng et al. 2006; C. Y. Peng 2008, pri-
vate communication). The two brightest spots in the
optical image are unresolved lensed images of the AGN
(henceforth ‘A’ and ‘B’,10also indicated as large crosses
in Figure 1). The optical emission is likely dominated
by the broad-line region of the AGN, and thus expected
to emerge from a compact circumnuclear pc-scale re-
gion.The quasar images have a brightness ratio of
b(B,A)=0.181. They coincide within 0.10′′with the po-
sitions measured by Keck at 430nm (2.2µm observed
frame; C03), i.e., within the relative astrometric er-
rors. These positions however are clearly offset from
the brightest peaks of the molecular line emission. The
third spot in the optical image is the lensing galaxy (‘G’,
also indicated as a small cross in Figure 1). The lensing
galaxy lies on the axis connecting images A and B, but
by more than a factor of 2 closer to image B. It however
lies in the very center of the molecular Einstein ring, as
expected. At the position of the lensing galaxy, we mea-
sure a 45.0GHz radio continuum flux of 79 ± 47µJy.
This corresponds to only 1.7σ, and thus has to be con-
sidered tentative at best.
In the adopted cosmology, the optical brightness of
PSSJ2322+1944 corresponds to an apparent bolomet-
ric luminosity of Lbol = 2.1 × 1014µ−1
et al. 2002). Assuming Eddington accretion, this cor-
responds to an apparent black hole mass of MBH =
7.0×109µ−1
lower limit, as black holes in z ∼ 4 quasars are found to
have typical accretion rates of˙ M = Lbol/LEdd= 0.3−0.4
(e.g., Shen et al. 2007).
In Figure 3, seven 42 kms−1wide velocity channels of
the CO(J=2→1) emission line are shown (channel 7 was
omitted from the data shown in Figures 1 and 2 due to
higher noise). Emission along the Einstein ring is de-
tected in all channels. Clearly, the emission is moving
systematically along the ring from the red part of the
CO(J=2→1) line to the blue. Note that there are many
peaks of similar surface brightness (peak fluxes of 450–
550µJybeam−1, or Tb=6–7K) that are found at different
positions at different velocities. This indicates, to first
order, a dynamical structure of uniform surface bright-
ness, where the components at different velocities get
projected to different positions in the lens plane. This
puts us in the unique situation to model the gravita-
tional lens configuration in this system in detail, and to
LM⊙ (where µL is
L,optL⊙ (Isaak
L,optM⊙. Note that this may be considered a
10By convention, A is the brightest image.
23 22 07.4007.3007.2007.1007.00
19 44 25
24
23
22
21
20
19
Fig. 2.— Contours of the CO(J=2→1) emission as shown in
Figure 1, but convolved to a linear resolution of 0.5′′and overlaid
on a HST NICMOS-2 image at λobs = 1.6µm (position crosses
in Figure 1). Contours are shown at (-3, -2, 2, 3, 4, 5, 6, 7, 8)×σ
(1σ = 49µJy beam−1). Note that the HST image was not cleaned,
and thus shows Airy rings around the unresolved quasar images
(Peng et al. 2006; C. Y. Peng 2008, private communication).
recover the intrinsic dynamical structure of the quasar
host galaxy in the background.
4. GRAVITATIONAL LENS INVERSION
In a previous attempt to model the gravitational
lensing effect toward the molecular gas reservoir of
PSSJ2322+1944, C03 adopted a generic model based on
the strongly lensed z=0.84 radio galaxy MG1131+0456.
By comparing various source configurations in the as-
sumed lens potential and based on plausibility argu-
ments, a model was found that described the overall
properties of the observed ring structure. To enable us
to describe the intrinsic properties of PSSJ2322+1944 in
more detail, we here present a direct model reconstruc-
tion and inversion of the lensing effect in this system.
This modeling is based on the new CO observations pre-
sented in the previous section.
4.1. Method: Bayesian Inference
Due to the remaining observational uncertainties, it
is not possible to derive a unique solution for the in-
version of the gravitational lens. We thus explored the
parameter space permitted by the data (and our mod-
eling assumptions) to find the best possible solution; if
some property of the models is consistent throughout this
permitted volume, it can be considered well-constrained
at high confidence. This parameter study thus follows
a Bayesian approach (Gregory 2005), using the Markov
Chain Monte Carlo (MCMC) code by Brewer & Lewis
(2006a). This algorithm does not simply aim at mini-
mizing χ2to find the best model (which may “overfit”
the data by fitting part of the noise), but explores the
whole range of plausible fits. By doing this, we can use
a large number of parameters (e.g. source pixel values)
in our model without overfitting the noise (since a broad
region in parameter space with higher χ2can outweigh
a particular solution that has very low χ2if the volume
of the parameter space near the very low χ2solution is

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4 Riechers et al.
1234
23 22 07.407.2 07.0
19 44 25
24
23
22
21
20
19
567
Fig. 3.— Channel maps of the CO(J=2→1) emission at 0.5′′resolution (beam size is shown in the bottom left corner of the first panel).
The same region is shown as in Fig. 2. One channel width is 6.25MHz, or 42 kms−1[frequencies increase with channel number and are
shown at 45016.35, 45022.60, 45028.85, 45035.10, 45041.35, 45047.60, and 45053.85MHz (red to blue)]. Contours are shown at (–3, –2, 2, 3,
4, 5, 6)×σ (1σ = 80µJybeam−1). Note that the noise in channel 7 is intrinsically higher by a factor of
very small).
The Bayesian analysis thus encodes the phase space of
possible source distributions allowed by the data into a
probability distribution (e.g., Gregory 2005). The lensed
image of the source was reconstructed based on the in-
tegrated CO emission line map and the structure de-
tected in the velocity channels, and then used to derive
a common model of the lensing galaxy’s projected den-
sity profile that reproduces the emission in all channels
simultaneously. Due to the differential structure among
the velocity channels, this implies that the reconstructed
source components after lens inversion will be different
for each velocity channel, and will reproduce the velocity
gradient across the source.
In this model description, the unknowns to be inferred
from the data are the seven source profiles {si}7
for each velocity channel, where si is shorthand for a
large number of (unknown) pixel values. The unknown
lens model parameters are denoted collectively by L.
Given observed data D, the posterior distribution for
the unknown parameters is proportional to the product
of the prior distribution and the likelihood function:
p({si},L|D) ∝ p({si},L)p(D|{si},L).
Here, D consists of seven extended images (one for each
velocity channel). We make the following standard as-
sumption for p(D|{si},L): Assuming that we know the
source and lens properties, we would predict the observed
image to be the lensed, blurred image of that source, plus
additive Gaussian noise:
?
2
i=1
where
Npixels
?
√2 relative to the other channels.
i=1, one
(1)
p(D|{si},L) ∝ exp −1
7
?
χ2
i
?
,(2)
χ2
i=
j=1
?Di,j− Mj(si,L)
σi
?2
.(3)
Here, Di,jis the j’th pixel of the i’th image, and M(si,L)
is the model image calculated by lensing and blurring the
i’th source with the proposed lens model L, and σiis the
noise standard deviation in the i’th image estimated from
the outer ‘blank sky’ regions of each image, i.e., distant
from any detected structure.
In this study, the lens was parameterized as a singular
isothermal elliptical potential with five free parameters,
which are the strength b of the lens, the ellipticity q of
the potential, the central position (xc,yc) of the lensing
source, and the angle of orientation θ of the projected
density profile. For this model, the lensing potential is
?
where
?x′
The source plane position (xs,ys) corresponding to any
lens plane position (x,y) is then given by the lens equa-
tion:
xs= x −∂φ
ys= y −∂φ
The optical position of the lensing galaxy was not used
as an initial model constraint to allow for a conservative
treatment of the errors. However, the optical data is used
in a second step to better constrain some of the source’s
intrinsic properties, as described in more detail below.
φ(x,y) = bqx′2+ y′2/q,(4)
y′
?
=
?
cosθ
−sinθ cosθ
sinθ
??x − xc
y − yc
?
.(5)
∂x,
∂y.
(6)
4.2. Application to Interferometric Data
Interferometer maps are reconstructed from visibility
data using a scale that samples one synthesized interfer-
ometer beam (i.e., resolution element) with multiple pix-
els. This means that the noise is not independent for all

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High–Resolution CO(2–1) Imaging of PSSJ2322+19445
pixels, as assumed by Equation 3. Correct modelling of
correlated noise is computationally very expensive, and
makes the evaluation of the likelihood (Equation 2) very
slow. In interferometric images, the scale of the syn-
thesized beam, or point-spread function (PSF), usually
is the same as the noise correlation scale (depending on
the interferometer baseline weighting function used in
the imaging Fourier transform). A good PSF model thus
will allow for a proper, but faster treatment of the noise
properties.
The correlation length scale of the noise was measured
at a clear distance from the detected molecular struc-
ture and used as the length scale for constructing an
optimized Gaussian PSF. This length scale predicts that
1/62 of the pixels are effectively independent at the scale
of the images (0.03′′pixel−1). This corresponds to 106
pixels on the scale of the Einstein ring. To not “overfit”
the image due to noise, only this fraction of informa-
tion can be used to computationally determine the lens
properties. This means that MCMC calculations have
to be run at an “annealing temperature” of 62, or alter-
natively, the σ’s can be artificially increased by a factor
√62 (e.g., Gregory 2005). The validity of this short-cut
procedure was confirmed by ensuring that the residuals
of the model images have the same statistical properties
as the background in the observed maps.
4.3. Priors
The prior distribution for the lens parameters is chosen
to be diffuse, but the final result is independent from this
selection. For the seven unknown sources representing
the source distribution in the velocity channels, we use
independent “massive inference” priors (Brewer & Lewis
2006b). To generate a random source, a moderate num-
ber of “atoms” of a certain brightness are added to a
blank source plane. These atoms are given a uniform
probability distribution in position and an exponential
probability distribution in flux. The width of each atom
in pixels is chosen at random from one of three values
to allow for the expectation that pixel values should be
correlated with their neighbours. This procedure gener-
ates a random source in which most pixels are dim, and
is a more appropriate prior for astronomical sources than
most conventional regularizers (Brewer & Lewis 2006a).
4.4. Modeling Results
4.4.1. Lens Parameters
From the MCMC run, we find that the lens has a
strength b = 0.745′′± 0.014′′, an ellipticity q = 0.969 ±
0.014, and a position (xc= 0.074′′± 0.024′′; yc= –0.109′′
± 0.029′′) [coordinates are relative to the center of the
model images at α=23h22m07s.176, δ=+19◦44′22′′.16].
Due to the fact that q ≃ 1, the lens potential is close
to circular, as is the projected lens density profile. Even
though the observed position of the lens (xobs
yobs
c
= –0.080′′) was not taken as an input parameter,
the model naturally reproduces its position within the
errors. The posterior probability distribution for θ is bi-
modal; with a probability of 62% (38%), the angle of
orientation of the potential is θ = 109.2◦± 7.1◦(62.7◦
± 6.5◦). The lens parameters are constrained well by
the molecular data alone, so the marginal posterior dis-
tributions are close to Gaussian (except for the bimodal
c
= 0.105′′;
θ distribution, which is well approximated by a mixture
of two Gaussians). Thus, all of the estimates and uncer-
tainties quoted above are of the form (mean ± standard
deviation).
The lensing model described above was derived based
on the distribution of the lensed CO emission only to
allow for a conservative treatment of the errors, and to
avoid a main systematic source of error: the remaining
astrometric uncertainties between the optical and radio
reference frames. The results of our study indicate that
the optical position of the lens is reproduced well, and
thus, that the astrometric offset appears to be small. Due
to the fact that gravitational lensing is achromatic, we
thus can use the model derived based on the distribu-
tion of the lensed CO emission only to also constrain the
intrinsic optical properties of the source. We also can
use the positions of the lensed images of the quasar in
the optical to further constrain the allowed parameter
space, and to estimate the position of the AGN within
the deprojected molecular gas reservoir.
Based on the sample of lenses that fit the molecular line
data, the optical positions of the quasar were ray-traced
back into the source plane. As the optical emission of
the source is compact, only those models that map both
quasar images onto the same position in the source plane
within the errors can be considered valid. We thus dis-
carded all models that did not fulfill this extra criterion.
We find that this extra constraint only marginally
changes previous results for the strength, ellipticity, and
position of the lens. However, it does impact the so-
lution for the angle of orientation of the potential, and
supports θ ∼ 109◦. As the source reconstructions pre-
sented in this section are not strongly sensitive to the
extra constraint, only results (and the more conservative
errors implied) from simulations produced without using
the optical data are shown unless stated otherwise.
4.4.2. Source Profiles
For the seven source profiles in the different velocity
channels, a “best” estimate was obtained by taking the
average of all sources encountered by the MCMC run.
This gives the posterior mean for each source, which is an
optimal estimate, as it minimizes the expected squared
error. As no unique solution exists for the lens inversion,
the full source sample has a certain diversity. However,
by taking the mean, only reproducible features are re-
tained from the full sample. Due to the fact that both the
resolution and sensitivity of the observations are finite,
the resulting surface brightness profiles are expected to
be smooth on a certain critical scale. Substructure may
appear on smaller scales, but will be smoothed out in
the final model due to the larger uncertainites involved
in reproducing such small structures across the permit-
ted volume in parameter space.
In Figure 4, the model-reconstructed (posterior mean)
molecular gas distribution in the seven CO velocity chan-
nels (columns) are shown in the lens plane (middle row),
and after lens inversion (top row), together with the ob-
servations (not convolved, i.e., at full spatial resolution;
bottom row). Since the lens model is common to all veloc-
ity channels, the changing distribution of molecular gas
between the velocity channels is due to intrinsic velocity
structure in the background source (note the difference
in scale between the lensed and the unlensed source).