11institutetext: Aalto University Metsähovi Radio Observatory, Metsähovintie
114, FI-02540 Kylmälä, Finland 22institutetext: Aalto University Department of
Electronics and Nanoengineering, P.O. Box 15500, FI-00076 AALTO, Finland
33institutetext: European Space Agency (ESA), European Space Astronomy Centre
(ESAC), Camino Bajo del Castillo s/n, 28692 Villanueva de la Cañada, Madrid,
Spain 44institutetext: Homer L. Dodge Department of Physics and Astronomy, The
University of Oklahoma, 440 W. Brooks St., Norman, OK 73019, USA
55institutetext: Dipartimento di Fisica e Astronomia ”G. Galilei”, Università
di Padova, Vicolo dell’Osservatorio 3, 35122 Padova, Italy 66institutetext:
European Southern Observatory (ESO), Alonso de Córdova 3107, Casilla 19,
Santiago 19001, Chile 77institutetext: Las Campanas Observatory, Carnegie
Observatories, Colina El Pino, Casilla 601, La Serena, Chile

# A host galaxy study of southern narrow-line Seyfert 1 galaxies

I. Varglund<EMAIL_ADDRESS>E. Järvelä Dodge Family Prize Fellow
in The University of Oklahoma3344 S. Ciroi 55 M. Berton 66 E. Congiu 66 A.
Lähteenmäki 1122 F. Di Mille 77

(Received )

We studied seven nearby narrow-line Seyfert 1 (NLS1) galaxies in $J$\- and
$Ks$-bands with redshifts varying from 0.019 to 0.092. This is the first
multi-source study targeting the hosts of southern NLS1 galaxies. Our data was
obtained with the FourStar instrument of the 6.5 m Magellan Baade telescope at
the Las Campanas Observatory (Chile). The aim of our study is to determine the
host galaxy morphologies of these sources by using GALFIT. We were able to
model six out of the seven sources reliably. Our conclusion is that all of the
reliably modelled sources are disk-like galaxies, either spirals or
lenticulars. None of these sources present elliptical morphology. Our findings
are in agreement with the hypothesis that disk-like galaxies are the main host
of jetted NLS1 galaxies. Taking advantage of observations in two bands, we
also produced a $J-Ks$ colour map of each source. Five of the six colour maps
show significant dust extinction near the core of the galaxy – a feature often
seen in gamma-ray-detected jetted NLS1 galaxies, and interpreted to be a
consequence of a past minor merger.

###### Key Words.:

galaxies: active – galaxies: Seyfert – galaxies: structure – infrared:
galaxies

##  1 Introduction

Table 1: Basic properties of the sample and the observations.

Source | Short name | $z$ | Scale | RA | Dec | Exp ($J$) | Exp ($Ks$) | Date | Seeing
---|---|---|---|---|---|---|---|---|---
|  |  | (kpc$/$″) | (J2000.0) | (J2000.0) | (s) | (s) | (year/month/day) | (″)
6dFGS J0622335-231742 | J0622-2317 | 0.019 | 0.370 | 06:22:34 | -23:17:42 | 960 | 384 | 2020/02/08 | 0.7
6dFGS J0952191-013644 | J0952-0136 | 0.037 | 0.705 | 09:52:19 | -01:36:44 | 583 | 384 | 2020/02/08 | 1.0
6dFGS J1511598-211902 | J1511-2119 | 0.044 | 0.832 | 15:12:00 | -21:19:02 | 640 | 384 | 2019/07/19 | 0.8
6dFGS J1522287-064441 | J1522-0644 | 0.083 | 1.500 | 15:22:29 | -06:44:41 | 640 | 384 | 2019/07/20 | 0.7
6dFGS J1615191-093613 | J1615-0936 | 0.064 | 1.182 | 16:15:19 | -09:36:13 | 640 | 384 | 2019/07/20 | 0.7
6dFGS J1628484-030408 | J1628-0304 | 0.092 | 1.645 | 16:28:48 | -03:04:08 | 640 | 384 | 2019/07/20 | 0.6
6dFGS J1646104-112404 | J1646-1124 | 0.074 | 1.351 | 16:46:14 | -11:24:04 | 872 | 523 | 2019/07/20 | 0.7

111Columns: (1) Source name, (2) Short name of the source, (3) Redshift, (4)
Scale at the redshift of the source, (5) Right ascension, (6) Declination, (7)
Total exposure time in $J$-band, (8) Total exposure time in $Ks$-band, (9)
Observation dates, (10) Average seeing during the observations measured from
the images in the band used for modelling.

Narrow-line Seyfert 1 (NLS1) galaxies (Osterbrock & Pogge 1985) are a class of
peculiar active galactic nuclei (AGN). The full width at half maximum (FWHM)
of H$\beta$ is by definition less than 2000 km s-1 (Goodrich 1989). This class
is often characterised by its unique optical spectral line features. One of
the most common identifiers of NLS1 galaxies is the faint [O III] emission
with respect to H$\beta$ ($\rm S([O\leavevmode\nobreak\ \textsc{III}]\lambda
5007)/S(H\beta)<3$, Osterbrock & Pogge 1985). Though not all NLS1 galaxies, a
considerable fraction of them exhibit strong Fe II multiplets (Boroson & Green
1992). Some NLS1 galaxies show blueshifted line profiles in high-ionization
lines, indicating outflows (e.g. Zamanov et al. 2002; Boroson 2005; Komossa et
al. 2008; Berton & Järvelä 2021).

NLS1 galaxies are intrinsically different from broad-line Seyfert (BLS1)
galaxies, for example, in their black hole masses, Eddington ratios, and their
multiwavelength and variability properties. The general consensus is that,
unlike BLS1 galaxies, NLS1 galaxies are powered by low-mass central
supermassive black holes ($M_{\text{BH}}\sim 10^{6}M_{\sun}$ \-
$10^{8}M_{\sun}$; Peterson 2011; Järvelä et al. 2015; Cracco et al. 2016; Chen
et al. 2018, e.g.). This has been confirmed, for example, by reverberation
mapping campaigns (e.g. Wang et al. 2016; Du et al. 2018; Du & Wang 2019). Due
to the low black hole mass, NLS1 galaxies are thought to be in an early
evolutionary stage with respect to BLS1 galaxies and other classes of AGN
(Mathur 2000). They also exhibit high Eddington ratios, generally between 0.1
and 1, but in some cases also exceeding unity (Boroson & Green 1992; Marziani
et al. 2018; Tortosa et al. 2022).

NLS1 galaxies have previously been thought to not have significant radio
emission (Komossa et al. 2006; Järvelä et al. 2015). Of the currently
identified NLS1 galaxies, roughly one sixth, $\sim 16\%$, have been detected
at radio frequencies: $\sim 6\%$ are weak radio emitters, and $\sim 10\%$ have
significant radio emission (Komossa et al. 2006). The rest of the NLS1
galaxies, $\sim 84\%$, have not been detected at radio frequencies, yet.

Finding disk galaxies, such as BLS1 galaxies, with jets is not uncommon (Keel
et al. 2006; Bagchi et al. 2014; Wu et al. 2022; Gao et al. 2023), however,
these sources in general have stellar masses similar to those of elliptical
galaxies and are thus atypical among disk-like galaxies. NLS1 galaxies, on the
other hand, have pseudo-bulges and low Sérsic indices, indicating that their
stellar masses are low (Sánchez Almeida 2020). Thus the discovery of
relativistic jets in some NLS1 galaxies was unexpected. These jets appear
similar, but less powerful, to those in blazars (e.g. Komossa et al. 2006;
Zhou et al. 2006; Yuan et al. 2008; Foschini 2011; Foschini et al. 2015;
Lähteenmäki et al. 2017, 2018). Typically, relativistic jets manifest strongly
in radio and dominate the whole band. However, also examples of jetted sources
with extremely weak low-frequency ($<$ 10 GHz) radio emission have been found
(Järvelä 2018; Berton et al. 2020; Järvelä et al. 2021; Sbarrato et al. 2021,
Järvelä et al. in prep.), questioning the utilisation of the emission in this
regime as a proxy for jettedness. The opposite has also been observed, when
sources with significant radio emission at low frequencies do not host
relativistic jets, even though that would be expected (Järvelä et al. 2022).
This anomaly is caused by the presence of significant star formation activity
(Caccianiga et al. 2015; Ganci et al. 2019; Berton et al. 2020).

There are several individual source analysis studies (Antón et al. 2008; León
Tavares et al. 2014; Kotilainen et al. 2016; D’Ammando et al. 2017; Olguín-
Iglesias et al. 2017; D’Ammando et al. 2018; Berton et al. 2019; Hamilton et
al. 2021; Vietri et al. 2022), as well as numerous large sample studies of the
hosts of NLS1 nuclei (Krongold et al. 2001; Crenshaw et al. 2003; Ohta et al.
2007; Orban de Xivry et al. 2011; Mathur et al. 2012; León Tavares et al.
2014; Järvelä et al. 2017; Olguín-Iglesias et al. 2020; Varglund et al. 2022).
Nearly all of these studies have identified them as disk-like. Elliptical
hosts have been an anomaly, however, some cases seem to exist (D’Ammando et
al. 2017, 2018).

In this paper we examined the hosts of seven southern NLS1 galaxies that are
also part of an optical spectral study and a multi-frequency radio imaging
study. This is the first systematic host galaxy investigation of southern NLS1
galaxies, and will be expanded in the future to allow statistically
significant comparison of their radio and optical properties. Our goal was to
determine the morphology of the hosts to improve the statistical weight of the
recent findings that most NLS1 nuclei are hosted in disk-like galaxies.
Furthermore, we also studied the colour maps as they could tell us more about
the properties of the core, such as its dust content, and shed light on
possible merger history. Ultimately, NLS1 galaxies as early-stage AGN, and
conveniently residing in the local Universe, can elucidate the evolution of
AGN over cosmic time.

Throughout this paper we assume standard $\lambda$CDM cosmology with
parameters H0 = 73 km s-1 Mpc-1, $\Omega_{\text{matter}}$ = 0.27, and
$\Omega_{\text{vacuum}}$ = 0.73 (Spergel et al. 2007).

Table 2: Radio properties of the sample (F = flat, S = steep), and the black
hole mass estimates.

Source | $S_{\mathrm{1.4\leavevmode\nobreak\ GHz}}$ | $S_{\mathrm{5.5\leavevmode\nobreak\ GHz}}$ | log $M_{\text{BH}}$ |  |
---|---|---|---|---|---
J0622-2317 | 4.3 | 0.9 | 6.47 | F | Diffuse
J0952-0136 | 59.8 | 21.3 | 6.33 | S | Jetted
J1511-2119 | 46.9 | 18.2 | 6.63 | S | Jetted?
J1522-0644 | 14.4 | 5.5 | 6.66 | S | Compact
J1615-0936 |  | 0.7 | 6.94 | S | Jetted?
J1628-0304 |  | 0.1 | 6.72 | F | Compact
J1646-1124 | 38.3 | 10.8 | 7.07 | S | Compact
222All values and radio properties from Chen et al. (2020).333Columns: (1)
Source name, (2) 1.4 GHz FIRST maximum flux density level, (3) 5.5 GHz maximum
flux density, (4) Black hole mass, (5) Radio spectrum, and (6) Radio
morphology

##  2 Observations and data reduction

Our sample of seven sources was selected from Chen et al. (2018). Our
selection criteria were based on the sources’ radio properties and the
possible presence of jets (Chen et al. 2020). The sources chosen for this
study have all been detected at 5.5 GHz, with five of them also having been
detected at 1.4 GHz. We have presented the radio flux densities as well as
some basic characteristics of our earlier observations in Table 2. The radio
spectrum of five of the sources is steep, while the final two sources have a
flatter spectrum. Based on the results of Chen et al. (2020), J0952-0136,
J1511-2119, and J1615-0936 were expected to harbour relativistic jets.
J0952-0136 has been identified as a jetted source also by Doi et al. (2015).
To investigate the radio properties of the Chen et al. (2020) sample in more
detail, we have obtained JVLA observations in 15, 22, and 33 GHz (Berton et
al. in prep). All of our sources have a low redshift ($<0.1$) and were
observed with FourStar, mounted on the 6.5 m Magellan Telescope. Five sources
were observed in July, 2019, while the rest were observed in February, 2020.
One source, J0952-0136, was saturated and could not be modelled.

The field-of-view (FoV) of FourStar is 5.4’ $\times$ 5.4’, with a scale of
0.159″px-1. All of our sources were observed in both $J$\- and $Ks$-bands.
When observing in $J$-band, we used 4-point dithering for all sources but
J0622-2317 for which we used 6-point dithering due to its brightness. In the
$Ks$-band the dithering used was 6-point for all sources.

For data reduction, we used IRAF. We first created a flat field image for each
source for each respective band, to remove eventual signatures in the data
produced by the instrument. This flat was applied to each dithered image.
After creating the flat field, we combined the dithered images to produce one
image per each dithering sequence. Then we continued by creating a superflat
to remove the final effects of the background. As the result, the mean sky
value is essentially zero. To finalise the reduction, we aligned the created
images and combined them. The data reduction process was done for both
filters. After producing the final images, we determined the zero point
magnitude for each source in both bands using
2MASS444https://www.ipac.caltech.edu/2mass magnitudes.

##  3 Data analysis

To perform the photometric decomposition of the host we used the band with
better seeing (see Table 1), which in all cases of this study resulted to be
the $J$-band. $Ks$-band images were only used to create the colour maps. Our
goal was to determine the host galaxy morphology and the related model
parameters that best describe the source.

To perform the photometric decomposition, we followed the same procedure as
described in Varglund et al. (2022) with a few exceptions. The PSF model was
built primarily with DAOPHOT and secondarily through choosing a star from the
image. No separate PSF stars were observed for this project. Furthermore, our
cutout image sizes were either $\sim$ 100 $\times$ 100 px or $\sim$ 200
$\times$ 200 px. The size of the image was dependent on the cutout size around
the host galaxy we were modelling. The sky background parameter was estimated
through two methods: the method described in Varglund et al. (2022) and by
using a large annulus that encompassed roughly a fifth of our image. We made
sure that no source was inside the annulus, before determining the sky
background value. The former method was used to confirm the results of the
latter method. Due to the used data reduction process, the sky background
value was virtually zero.

The fitting process with GALFIT version 3 (Peng et al. 2010), creating the
radial surface plot with the IRAF (Stetson 1987) task $ELLIPSE$, and the error
estimation method was identical to the process in Varglund et al. (2022). The
radial surface plot was created to study both the accuracy of the fit as well
as to see how the brightness of the components changes as a function of the
distance from the core. On average the limiting surface brightness of the
images is around 22-23 mag arcsec-2 whereas all of our sources are
considerably brighter than the limit, and thus we can model them reliably up
to large radii.

In the case of a suspected bar-component based on the best-fit parameters we
used an additional diagnostic to determine if the component truly is a bar. To
do this, we extracted the position angle (PA) and ellipticity parameters of
the isophotes obtained from the ELLIPSE output of the galaxy image. These
parameters were plotted against the radius. The plot indicates the presence of
a bar if there is a region in which the ellipticity increases while the PA
remains mostly constant ($\Delta$PA $<20$°) (Wozniak et al. 1995; Menéndez-
Delmestre et al. 2007).

Finally, since we had images from two bands for each source, we were able to
create colour maps of all of our sources. To obtain the colour map, we used
the equation below:

$\texttt{Colour image pixel
value}=(\text{zp}_{J}-\text{zp}_{Ks})-2.5\Bigg{[}\text{log}_{10}\bigg{(}\frac{J}{Ks}\bigg{)}\Bigg{]},$
(1)

The data used when calculating the colour image was corrected for the
different exposure times of the bands. The images were aligned by using
GEOMAP, GEOTRAN, and IMALIGN to guarantee that the images were perfectly
aligned. The data processing allowed for the PSFs to be nearly identical,
improving the colour maps. In the above equation, $zp$ stands for the zero
point magnitude, with the subscript defining the band in question. The contour
levels in the maps are defined using the Ks- band image sky standard deviation
(std) and are at 20, 40, 60, 80, 100, 180, 360, 720, and 1440 $\times$ std.

Table 3: Best-fit parameters of J0622-2317. $\chi^{2}_{\nu}$ = 0.988
$\begin{subarray}{c}+0.00\\\ -0.00\end{subarray}$.

Function | Mag | $r_{\text{e}}$ | $n$ | Axial | PA | Notes
---|---|---|---|---|---|---
|  | (kpc) |  | ratio | (°) |
PSF 1 | 14.70 $\begin{subarray}{c}+0.01\\\ -0.01\end{subarray}$ |  |  |  |  |
Sérsic 1 | 11.98 $\begin{subarray}{c}+0.01\\\ -0.02\end{subarray}$ | 3.20 $\begin{subarray}{c}+0.19\\\ -0.00\end{subarray}$ | 1.65 $\begin{subarray}{c}+0.01\\\ -0.00\end{subarray}$ | 0.80 $\begin{subarray}{c}+0.00\\\ -0.00\end{subarray}$ | 41.91 $\begin{subarray}{c}+0.01\\\ -0.02\end{subarray}$ | Disk and bulge
555Columns: (1) Function used for modelling, (2) Magnitude, (3) Effective
radius, (4) Sérsic index, (5) Axial ratio, (6) Position angle, (7) Additional
notes on the sources.

Figure 1: J-band images of J0622-2317. The FoV is 31.8″$/$ 11.8 kpc in all
images. _Left panel:_ observed image, _middle panel:_ model image, and _right
panel:_ residual image, smoothed over 3px.

Figure 2: Radial surface brightness profile plot of J0622-2317. The blue line
represents the galaxy component, the dashed red line shows the model, the
dotted green line marks the PSF, and finally the dashed pink line shows the
Sérsic component. The error of each component is shown by the shaded area
surrounding each curve. Figure 3: $J$ \- $Ks$ colour map of J0622-2317.

##  4 Individual source analysis

###  4.1 J0622-2317

According to Chen et al. (2020) this source has a flat in-band radio spectrum,
and a diffuse radio morphology. Additional information of this source and all
the sources of our sample can be found in Table 2. A visual inspection shows a
very clear spiral structure and even hints at the existence of a bar (Fig. 3).

The size of the fitting region was limited due to a nearby, very bright,
saturated star. The best-fit results can be seen in Table 5. In addition to
the NLS1, also three nearby sources were fitted using the PSF model to obtain
a more realistic $\chi^{2}_{\nu}$ for the FoV. We have most likely fitted a
disk, or a combination of a disk and a pseudo-bulge component that cannot be
separated and modelled individually. The bulge may be unresolved, but its
light is still present in the image, thus affecting the model, or the bulge is
either not bright enough or does not have different enough properties compared
to the disk. The component is small, however, the Sérsic index is high for a
disk so it is possible that a bulge component is included. The intensity plots
of the source can be seen in Fig. 3. There are clear residuals left due to the
spiral arms and some slight residuals near the core of the galaxy. We tried to
include a bar into the model, but were unable to obtain good results.

The radial surface plot can be seen in Fig 3. As expected, the model image
curve does not perfectly follow the curve for the galaxy image. This is most
likely due to the spiral arms not being completely modelled. The colour map
can be seen in Fig. 3. On average, Seyfert galaxies have a $J$ \- $Ks$
magnitude between 0 and 2.5, and when the magnitude difference is 1.3 or more,
a galaxy is considered red (Jarrett 2000). Unlike typical Seyfert galaxies,
J0622-2317 is blue. However, the magnitude of the core of the galaxy is above
0, and is thus within a common range found in NLS1 galaxies. As with the other
images, the spiral arms are clearly visible in the colour map.

###  4.2 J1511-2119

Table 4: Best-fit parameters of J1511-2119. $\chi^{2}_{\nu}$ = 0.916
$\begin{subarray}{c}+0.00\\\ -0.02\end{subarray}$.

Function | Mag | $r_{\text{e}}$ | $n$ | Axial | PA | Notes
---|---|---|---|---|---|---
|  | (kpc) |  | ratio | (°) |
PSF 1 | 14.24 $\begin{subarray}{c}+0.03\\\ -0.03\end{subarray}$ |  |  |  |  |
Sérsic | 15.90 $\begin{subarray}{c}+0.03\\\ -0.03\end{subarray}$ | 3.93 $\begin{subarray}{c}+0.15\\\ -0.00\end{subarray}$ | 2.96 $\begin{subarray}{c}+0.08\\\ -0.00\end{subarray}$ | 0.90 $\begin{subarray}{c}+0.00\\\ -0.00\end{subarray}$ | -70.82 $\begin{subarray}{c}+0.00\\\ -0.12\end{subarray}$ | Bulge and disk
666Columns: (1) Function used for modelling, (2) Magnitude, (3) Effective
radius, (4) Sérsic index, (5) Axial ratio, (6) Position angle, (7) Additional
notes on the sources.

Figure 5: Radial surface brightness profile plot of J1511-2119. The blue line
represents the galaxy component, the dashed red line shows the model, the
dotted green line marks the PSF, the dashed pink line shows the Sérsic 1
component, the final component, Sérsic 2, is shown with dotted light blue. The
error of each component is shown by the shaded area surrounding each curve.
Figure 6: $J$ \- $Ks$ colour map of J1511-2119.

Figure 4: J-band images of J1511-2119. The FoV is 31.8″$/$ 26.5 kpc in all
images. _Left panel:_ observed image, _middle panel:_ model image, and _right
panel:_ residual image, smoothed over 3px.

This is one of the three jetted sources in this study. Its radio map can be
found in Chen et al. (2020). We used a DAOPHOT PSF model when fitting. There
were ten nearby sources in our FoV, all of which we were able to fit with a
single PSF with very little residuals.

Our fit models a combination of a disk and a bulge (Table 4). The Sérsic index
of 2.96 does not strongly support any specific morphology, even though indices
this high can be found also in spiral galaxies (Elmegreen & Elmegreen 2014).
The size of the galaxy is typical for an NLS1 host. The excess residuals (Fig.
6) near the core imply the presence of unmodelled structures, such as leftover
fragments of the bulge. The galaxy, model, and Sérsic curves in the radial
surface brightness plot, Fig. 6, are all very similar to each other.

The host galaxy of J1511-2119 is disk-like, and based on the Sérsic index, an
S0 galaxy. The colour map, shown in Fig. 6, shows colours commonly seen in
NLS1 galaxies, with the red most likely signifying dust extinction. The core
of the galaxy is extremely red as the $J$-$Ks$ magnitude is $\sim$2\. Beyond
the core, the galaxy has very uniform colours.

###  4.3 J1522-0644

We successfully used the PSF model to fit the five nearby sources. The source,
model, and residual images can be seen in Fig. 10. The best-fit parameters are
given in Table 7. Along with a PSF, we used two Sérsic functions for fitting
this source. Based on the effective radius of Sérsic 1 it is presumably
fitting a disk. The Sérsic index is low for a disk, but not unreasonable.
Based on its parameters, the second Sérsic function most likely models a bar.
To further study this possibility, we plotted the PA and ellipticity over the
radius, seen in Fig. 10. The PA remains steady throughout the plot. The
ellipticity increases toward the higher radii. Based on this plot, it is
difficult to confirm whether or not Sérsic 2 truly models a bar. The remaining
residuals of the source are mostly remnants of the spiral arms.

The $\chi^{2}_{\nu}$ is relatively high due to the remaining residuals, but
the fit is satisfactory and the radial surface plot, Fig. 10, supports the
goodness of the fit. The galaxy and the model curves match. The colour
magnitude, seen in Fig. 10, of the galaxy is as expected in an NLS1 galaxy.
The $J$-$Ks$ magnitude near the core is approximately 1.75, and is thus very
red. The redness of the core is most likely due to dust extinction.

Table 5: Best-fit parameters of J1522-0644. $\chi^{2}_{\nu}$ = 1.257
$\begin{subarray}{c}+0.03\\\ -0.03\end{subarray}$.

Function | Mag | $r_{\text{e}}$ | $n$ | Axial | PA | Notes
---|---|---|---|---|---|---
|  | (kpc) |  | ratio | (°) |
PSF 1 | 15.55 $\begin{subarray}{c}+0.05\\\ -0.05\end{subarray}$ |  |  |  |  |
Sérsic 1 | 14.76 $\begin{subarray}{c}+0.01\\\ -0.00\end{subarray}$ | 7.72 $\begin{subarray}{c}+0.02\\\ -0.02\end{subarray}$ | 0.48 $\begin{subarray}{c}+0.01\\\ -0.01\end{subarray}$ | 0.76 $\begin{subarray}{c}+0.00\\\ -0.00\end{subarray}$ | -5.97 $\begin{subarray}{c}+0.02\\\ -0.02\end{subarray}$ | Disk
Sérsic 2 | 15.19 $\begin{subarray}{c}+0.00\\\ -0.00\end{subarray}$ | 3.80 $\begin{subarray}{c}+0.00\\\ -0.00\end{subarray}$ | 0.61 $\begin{subarray}{c}+0.00\\\ -0.00\end{subarray}$ | 0.36 $\begin{subarray}{c}+0.00\\\ -0.00\end{subarray}$ | -44.74 $\begin{subarray}{c}+0.01\\\ -0.01\end{subarray}$ | Bar
777Columns: (1) Function used for modelling, (2) Magnitude, (3) Effective
radius, (4) Sérsic index, (5) Axial ratio, (6) Position angle, (7) Additional
notes on the sources.

Figure 7: J-band images of J1522-0644. The FoV is 31.8″$/$ 47.7 kpc in all
images. _Left panel:_ observed image, _middle panel:_ model image, and _right
panel:_ residual image, smoothed over 3px.

Figure 8: Radial surface brightness profile plot of J1522-0644. The blue line
represents the galaxy component, the dashed red line shows the model, the
dotted green line marks the PSF, and finally the dashed pink line shows the
Sérsic component. The error of each component is shown by the shaded area
surrounding each curve. Figure 9: $J$ \- $Ks$ colour map of J1522-0644. Figure
10: Position angle and ellipticity of the Sérsic 1 function plotted against
the major axis of the isophote of J1522-0644.

###  4.4 J1615-0936

We tried to model the PSF with DAOPHOT and by directly using a nearby star.
Based on our results, the DAOPHOT PSF provided a better model. The
unsatisfactory fit of the nearby source increases the $\chi^{2}_{\nu}$ of the
fit.

We modelled the source with a PSF, a Sérsic function, and an exponential disk
function. The best-fit parameters can be found in Table 8. Despite several
attempts and different approaches, the $\chi^{2}_{\nu}$ remained quite high.
Based on our best-fit results, the Sérsic represents a pseudo-bulge and the
exponential disk function naturally fits a disk. There are some clear
residuals left, as seen in Fig. 13. The residuals of the NLS1 are most likely
caused by the bright AGN that was hard to model due to the lack of a bright
enough PSF star.

The $\chi^{2}_{\nu}$ of the fit is not ideal, however, it does not always
imply that the fit is physically unreasonable (Dewsnap et al. 2022). Indeed,
based on the radial surface brightness plot, seen in Fig. 13, our model is
accurate, as there is virtually no deviation between the model and galaxy
curves, and the obtained components are physically reasonable. Based on our
results, the host galaxy of J1615-0936 is disk-like, resembling a lenticular
galaxy. This is supported by the axial ratio of the disk component being so
small, 0.29 $\begin{subarray}{c}+0.00\\\ -0.00\end{subarray}$.

The colour map of J1615-0936 is presented in Fig. 13. Based on it, the core
has a $J$-$Ks$ colour magnitude of roughly 1.5, which means that the core of
the galaxy is very red. The rest of the galaxy presents uniform colours.
Overall, the colours of the galaxy are as would be expected in an NLS1 galaxy.

Table 6: Best-fit parameters of J1615-0936. $\chi^{2}_{\nu}$ = 2.553
$\begin{subarray}{c}+0.02\\\ -0.01\end{subarray}$.

Function | Mag | $r_{\text{e}}$ | $n$ | Axial | PA | Notes
---|---|---|---|---|---|---
|  | (kpc) |  | ratio | (°) |
PSF 1 | 14.25 $\begin{subarray}{c}+0.03\\\ -0.03\end{subarray}$ |  |  |  |  |
Sérsic | 14.68 $\begin{subarray}{c}+0.03\\\ -0.03\end{subarray}$ | 1.61 $\begin{subarray}{c}+0.05\\\ -0.04\end{subarray}$ | 2.74 $\begin{subarray}{c}+0.15\\\ -0.12\end{subarray}$ | 0.72 $\begin{subarray}{c}+0.00\\\ -0.00\end{subarray}$ | 81.49 $\begin{subarray}{c}+0.07\\\ -0.05\end{subarray}$ | Bulge
Exp.disk | 17.62 $\begin{subarray}{c}+0.03\\\ -0.03\end{subarray}$ | 2.44 $\begin{subarray}{c}+0.00\\\ -0.03\end{subarray}$ |  | 0.29 $\begin{subarray}{c}+0.00\\\ -0.00\end{subarray}$ | 87.69 $\begin{subarray}{c}+0.00\\\ -0.00\end{subarray}$ | Disk
888Columns: (1) Function used for modelling, (2) Magnitude, (3) Effective
radius, (4) Sérsic index, (5) Axial ratio, (6) Position angle, (7) Additional
notes on the sources.

Figure 11: J-band images of J1615-0936. The FoV is 15.9″$/$ 18.8 kpc in all
images. _Left panel:_ observed image, _middle panel:_ model image, and _right
panel:_ residual image, smoothed over 3px.

Figure 12: Radial surface brightness profile plot of J1615-0936. The blue line
represents the galaxy component, the dashed red line shows the model, the
dotted green line marks the PSF, the dashed pink line shows the Sérsic
component, the final component, the exponential disk, is shown with dotted
light blue. The error of each component is shown by the shaded area
surrounding each curve. Figure 13: $J$ \- $Ks$ colour map of J1615-0936.

###  4.5 J1628-0304

We used a Sérsic function, an exponential disk function, and a PSF to fit the
galaxy. The best-fit parameters are shown in Table 9. DAOPHOT provided the
best PSF model, but it is not perfect, as evidenced by the remaining
residuals, seen in Fig. 17, near the core of the galaxy. The Sérsic function
models a bar component. The galaxy is small, also proven by the small
effective radius of the exponential disk component. We plotted the ellipticity
and PA against the radius of the isophotes, seen in Fig. 17, to study the
possible bar. In the plot the ellipticity can be seen increasing while the PA
is more or less constant. The plot strongly suggests that the Sérsic function
indeed models a bar component.

The $\chi^{2}_{\nu}$ for this source is quite high. However, there are
virtually no high surface brightness residuals left. The remaining residuals
are likely caused by the imperfect PSF model, which we could not improve
despite several attempts with different methods. A major issue with the PSF
modelling was the lack of bright enough stars in the FoV – none of them are
brighter than the AGN, and probably cannot model the wings of the PSF
accurately. However, as discussed earlier, the $\chi^{2}_{\nu}$ value does not
alone determine if the fit is physically justifiable and we want to emphasise
the importance of taking into account also the obtained components, the
residuals, and the comparison between the galaxy and and the model, in Fig.
17, when evaluating the fit. Based on all these aspects, the obtained results
are adequate. The host galaxy of J1628-0304 is disk-like.

The colour map can be seen in Fig. 17. The colours of this galaxy are
interesting as the reddest part is not at the core, but rather surrounds it.
The structure could be due to a star forming ring or a minor merger (Iodice et
al. 2014). However, the resolution of our images does not allow a detailed
view of the possible minor merger and due to this, the feature is most likely
a star forming ring. Circumnuclear star forming rings have been frequently
observed in NLS1 galaxies (Winkel et al. 2022). Nonetheless, the core has
colour magnitude of $\sim$1.5 and is thus very red.

Table 7: Best-fit parameters of J1628-0304. $\chi^{2}_{\nu}$ = 3.183
$\begin{subarray}{c}+0.05\\\ -0.05\end{subarray}$.

Function | Mag | $r_{\text{e}}$ | $n$ | Axial | PA | Notes
---|---|---|---|---|---|---
|  | (kpc) |  | ratio | (°) |
PSF 1 | 14.37 $\begin{subarray}{c}+0.16\\\ -0.17\end{subarray}$ |  |  |  |  |
Sérsic | 18.16 $\begin{subarray}{c}+0.18\\\ -0.17\end{subarray}$ | 1.92 $\begin{subarray}{c}+0.03\\\ -0.00\end{subarray}$ | 0.75 $\begin{subarray}{c}+0.00\\\ -0.00\end{subarray}$ | 0.44 $\begin{subarray}{c}+0.00\\\ -0.00\end{subarray}$ | 46.06 $\begin{subarray}{c}+0.15\\\ -0.00\end{subarray}$ | Bar
Exp. disk | 17.45 $\begin{subarray}{c}+0.16\\\ -0.17\end{subarray}$ | 2.00 $\begin{subarray}{c}+0.01\\\ -0.02\end{subarray}$ |  | 0.57 $\begin{subarray}{c}+0.01\\\ -0.00\end{subarray}$ | 43.23 $\begin{subarray}{c}+0.31\\\ -0.00\end{subarray}$ | Disk
999Columns: (1) Function used for modelling, (2) Magnitude, (3) Effective
radius, (4) Sérsic index, (5) Axial ratio, (6) Position angle, (7) Additional
notes on the sources.

Figure 14: J-band images of J1628-0304. The FoV is 15.9″$/$ 26.2 kpc in all
images. _Left panel:_ observed image, _middle panel:_ model image, and _right
panel:_ residual image, smoothed over 3px.

Figure 15: Radial surface brightness profile plot of J1628-0304. The blue line
represents the galaxy component, the dashed red line shows the model, the
dotted green line marks the PSF, the dashed pink line shows the Sérsic
component, the final component, the exponential disk component is shown with
dotted light blue. The error of each component is shown by the shaded area
surrounding each curve. Figure 16: $J$ \- $Ks$ colour map of J1628-0304.
Figure 17: Position angle and ellipticity of the Sérsic function plotted
against the major axis of the isophote of J1628-0304.

###  4.6 J1646-1124

This source has a high $\chi^{2}_{\nu}$ value (Table 8). Despite this, when
studying the residuals, seen in Fig. 20, and the best-fit parameters, the fit
is acceptable. A PSF was fitted on the five nearby sources as well as on the
galaxy. In addition, the galaxy required one Sérsic function, with the
function most likely modelling a bulge. The Sérsic index is high for a spiral
galaxy, though still within the range of accepted values for them. We
attempted fitting additional components, but could not achieve a physically
justifiable fit. Our issues mostly stem from the AGN being so bright. A fit
with three Sérsic functions yielded a better $\chi^{2}_{\nu}$ ( 2), but one of
the components consistently remained unresolved, likely trying to fit parts of
the point-like AGN. The resolved components modelled a bulge and a disk
component. The alternative-fit results can be seen in Table 9.

Some residuals still remain near the galaxy core, and are most likely due to
the issues with the PSF model. There is indication of a spiral-like structure
surrounding the galaxy, starting from its west side, and bending anti-
clockwise toward east. However, without further proof, the assessment of the
residuals remains a speculation. Due to the $\chi^{2}_{\nu}$ value, the
residuals, and the best-fit parameters, we do not deem it possible to
irrefutably determine the host galaxy of J1646-1124. However, there is
evidence, for example, in the alternative fit, that J1646-1124 is disk-like,
and based on the parameters, an S0.

The radial surface plot, seen in Fig. 20, shows that despite the difficulties
in the modelling, the model fits the galaxy very well up to the radius of 6
kpc. The colour map of the source can be seen in Fig. 20. The colours are as
expected for an NLS1 galaxy, with the core being very red ($\sim$2.2). The
redness is most likely due to dust extinction. Some slight saturation,
resulting in the polygonal shape, due to the $Ks$-image, can be seen in the
colour map.

Table 8: Best-fit parameters of J1646-1124. $\chi^{2}_{\nu}$ = 3.221
$\begin{subarray}{c}+0.02\\\ -0.01\end{subarray}$.

Function | Mag | $r_{\text{e}}$ | $n$ | Axial | PA | Notes
---|---|---|---|---|---|---
|  | (kpc) |  | ratio | (°) |
PSF 1 | 13.76 $\begin{subarray}{c}+0.00\\\ -0.00\end{subarray}$ |  |  |  |  |
Sérsic | 14.63 $\begin{subarray}{c}+0.05\\\ -0.06\end{subarray}$ | 1.65 $\begin{subarray}{c}+0.03\\\ -0.03\end{subarray}$ | 2.68 $\begin{subarray}{c}+0.15\\\ -0.12\end{subarray}$ | 0.81 $\begin{subarray}{c}+0.00\\\ -0.00\end{subarray}$ | -72.19 $\begin{subarray}{c}+0.03\\\ -0.05\end{subarray}$ | Bulge
101010Columns: (1) Function used for modelling, (2) Magnitude, (3) Effective
radius, (4) Sérsic index, (5) Axial ratio, (6) Position angle, (7) Additional
notes on the sources.

Table 9: Alternative-fit parameters of J1646-1124. $\chi^{2}_{\nu}$ = 2.229
$\begin{subarray}{c}+0.03\\\ -0.04\end{subarray}$.

Function | Mag | $r_{\text{e}}$ | $n$ | Axial | PA | Notes
---|---|---|---|---|---|---
|  | (kpc) |  | ratio | (°) |
PSF 1 | 13.83 $\begin{subarray}{c}+0.00\\\ -0.00\end{subarray}$ |  |  |  |  |
Sérsic 1 | 15.94 $\begin{subarray}{c}+0.05\\\ -0.06\end{subarray}$ | 1.88 $\begin{subarray}{c}+0.02\\\ -0.04\end{subarray}$ | 1.38 $\begin{subarray}{c}+0.04\\\ -0.21\end{subarray}$ | 0.72 $\begin{subarray}{c}+0.00\\\ -0.00\end{subarray}$ | 66.76 $\begin{subarray}{c}+0.03\\\ -0.05\end{subarray}$ | Disk
Sérsic 2 | 15.78 $\begin{subarray}{c}+0.04\\\ -0.03\end{subarray}$ | 0.37 $\begin{subarray}{c}+0.02\\\ -0.03\end{subarray}$ | 0.34 $\begin{subarray}{c}+0.04\\\ -0.02\end{subarray}$ | 0.17 $\begin{subarray}{c}+0.00\\\ -0.00\end{subarray}$ | 13.43 $\begin{subarray}{c}+0.08\\\ -0.05\end{subarray}$ |
Sérsic 3 | 15.23 $\begin{subarray}{c}+0.05\\\ -0.06\end{subarray}$ | 3.11 $\begin{subarray}{c}+0.24\\\ -0.13\end{subarray}$ | 2.52 $\begin{subarray}{c}+0.11\\\ -0.12\end{subarray}$ | 0.70 $\begin{subarray}{c}+0.00\\\ -0.00\end{subarray}$ | -89.6 $\begin{subarray}{c}+0.01\\\ -0.05\end{subarray}$ | Bulge
111111Columns: (1) Function used for modelling, (2) Magnitude, (3) Effective
radius, (4) Sérsic index, (5) Axial ratio, (6) Position angle, (7) Additional
notes on the sources.

Figure 18: J-band images of J1646-1124. The FoV is 15.9″$/$ 21.5 kpc in all
images. _Left panel:_ observed image, _middle panel:_ model image, and _right
panel:_ residual image, smoothed over 3px.

Figure 19: Radial surface brightness profile plot of J1646-1124. The blue line
represents the galaxy component, the dashed red line shows the model, the
dotted green line marks the PSF, the dashed pink line shows the Sérsic 1
component, the final component, Sérsic 2, is shown with dotted light blue. The
error of each component is shown by the shaded area surrounding each curve.
Figure 20: $J$ \- $Ks$ colour map of J1646-1124.

##  5 Discussion

We were able to model six out of the seven sources in our sample. The only
source we could not model was J0952-0136. A summary with the final verdicts of
each source can be found in Table 10. Two of our modelled sources, J1511-2119
and J1615-0936, have been speculated by Chen et al. (2020) to be jetted based
on their radio spectra. All six of our sources have disk-like hosts. The only
uncertain host morphology is for J1646-1124, however, our results still
suggest the possibility of disk-like morphology. Three of our sources show
evidence for being in lenticular galaxies. However, it is unclear if the
morphology truly is lenticular or if the results are caused by the AGN being
too bright to have a reliable PSF model from the FoV. We were able to model a
disk component in five of our sources, which naturally indicates disk-like
morphology of the galaxy. A bar component can be found in two sources,
J1522-0644 and J1628-0304. Based on the PA and ellipticity versus the major
axis of the isophote plots, J1628-0304 is very likely to host a bar. We were
able to model a bulge component in four sources. The modelled bulges have a
slightly high Sérsic index to be considered pseudo-bulges in disk-like
galaxies. The commonly used limit for distinguishing between pseudo- and
classical bulges is $n=2$, where indices below 2 are associated with pseudo-
bulges and indices above with classical bulges (Orban de Xivry et al. 2011).
However, the sources that have bulges with Sérsic indices above 2 are the
sources we suspect of being S0s and it has been suggested that S0/a sources
can have pseudo-bulges with Sérsic index $\approx 2.4$ with a standard
deviation of 0.66 (Kormendy & Kennicutt 2004). With this in mind, the modelled
bulges are most-likely pseudo-bulges, which is in agreement with what is seen
in disk-like galaxies.

Our findings are in alignment with the majority of previous studies regarding
the host galaxy morphology of jetted and non-jetted NLS1 galaxies (e.g.
Crenshaw et al. 2003; Ohta et al. 2007; Antón et al. 2008; Orban de Xivry et
al. 2011; León Tavares et al. 2014; Kotilainen et al. 2016; Olguín-Iglesias et
al. 2017; Järvelä et al. 2018; Berton et al. 2019; Olguín-Iglesias et al.
2020; Hamilton et al. 2021; Vietri et al. 2022; Varglund et al. 2022). Only
two jetted NLS1 galaxies have so far been identified to be located in
elliptical galaxies (D’Ammando et al. 2017, 2018). However, the morphology of
FBQS J1644+2619 being elliptical is up for debate as the source has also been
suggested to reside in a barred lenticular galaxy.

Only a few studies on NLS1 galaxy stellar masses exist, and we were able to
find a stellar mass estimate for three of our sources: J0952-0136
($M_{\text{star}}\sim 1.5\times 10^{9}M_{\sun}$) (Pan et al. 2021), J0622-2317
($M_{\text{star}}\sim 1.3\times 10^{10}M_{\sun}$) (Saulder et al. 2016), and
J1511-2119 ($M_{\text{star}}\sim 1.3\times 10^{11}M_{\sun}$) (Koss et al.
2021). Two of these sources have a low stellar mass, while J1511-2119 has a
high stellar mass. The high stellar mass is not surprising as the galaxy is
most likely an S0. The stellar mass of J0952-0136 being low is interesting as
it is a jetted source, however, as we do not have information of the stellar
mass of other jetted NLS1 galaxies, no further conclusions can be drawn. Based
on the relation of the stellar mass, and the effective radius and Sérsic index
of the NLS1 galaxies of this paper, the stellar mass of these sources is most
likely low (Sánchez Almeida 2020).

When studying the evolution of AGN, the nearby environment, both group- and
galaxy-scale, is a crucial factor to take into account. This is due to the
nearby environment being capable of reshaping the galaxy as well as the galaxy
dynamics. It has been hypothesised that jetted NLS1 galaxies and
interaction/mergers might be linked (e.g. Järvelä et al. 2017; Olguín-Iglesias
et al. 2020; Berton et al. 2021). However, there is no clear consensus on the
topic yet. None of the sources in this paper present clear mergers or
interaction. This could be for a variety of reasons, one of the possible
reasons being the lack of resolution for seeing minor mergers. It has been
shown before that minor mergers are capable of triggering nuclear activity, an
example of this is IRAS 17020+4544. Although this source is a jetted NLS1
galaxy, there is no clear evidence of mergers in optical/NIR imaging, but
there is clear evidence of a small companion galaxy based on molecular gas
observations (Salomé et al. 2021; Järvelä et al. 2022).

It has also been speculated that $\gamma$-ray-emitting NLS1 galaxies have
experienced more minor mergers than non-$\gamma$-NLS1 galaxies with
significant radio emission (Olguín-Iglesias et al. 2020). The same study also
stated that the $\gamma$-NLS1 galaxies have bulge colours that are
significantly different from their hosts with red NIR colour, $J-K$ $\sim$ 2,
while the non-$\gamma$-ray NLS1 galaxies have bulge colours that are similar
to the rest of the galaxy. The red nuclei is, according to the authors, most
likely due to either the bulge showing enhanced star formation or the AGN
being dust-embedded as a consequence of a past minor merger. Contradictory to
the results in Olguín-Iglesias et al. (2020), all but one of the sources in
this paper showcase the nucleus being significantly more red than the rest of
the galaxy. The only exception to this is J0622-2317, however, it is possible
that the colouration difference in this source is due to the spiral arms
beginning directly from the core, thus ”hiding” the core under the spiral
arms. In case the speculation by Olguín-Iglesias et al. (2020) is indeed
correct, it is entirely possible that the colour of the core could be a hint
of previous minor mergers also in our sources. However, unlike in Olguín-
Iglesias et al. (2020), the nuclei of our sources do not present any clear
difference in colours between jetted and non-jetted NLS1 galaxies. Based on
the current understanding, it is impossible to say whether or not mergers are
needed for triggering powerful relativistic jets in NLS1 galaxies.

Table 10: Summary of the results. Columns: (1) Source name, (2) Probable morphology of the host galaxy, (3) Components of the model, (4) Presence of a relativistic jet based on the Metsähovi 37 GHz detections, (5) Large-scale environment parameter, (6) Additional notes. Source | Morphology | Components | Jetted | Notes
---|---|---|---|---
6dFGS J0622335-231742 | Disk-like | Disk and bulge | No |
6dFGS J0952191-013644 | Unclear |  | Yes | Saturated image
6dFGS J1511598-211902 | Disk-like | Disk and bulge | Yes | Possibly S0
6dFGS J1522287-064441 | Disk-like | Disk and bar | No |
6dFGS J1615191-093613 | Disk-like | Disk and bulge | Yes | Possibly S0
6dFGS J1628484-030408 | Disk-like | Disk and bar | No |
6dFGS J1646104-112404 | Disk-like | Bulge | No | Slightly saturated colour map, possibly S0

##  6 Conclusions

We have studied six NLS1 galaxies that have been observed in $J-$ and
$Ks$-bands. The observations were carried out with the FourStar NIR
instrument. The goal of our study was to investigate what morphology the host
galaxies of our sources have. We also created colour maps for all of our
sources. These maps were created to help study the properties of the core of
the source. Furthermore, we also wanted to study possible interaction in our
sources. The main findings of our study are:

  1. 1.

All of our sources are most likely located in disk-like galaxies, including
the possibly jetted NLS1 galaxies.

  2. 2.

The colour maps show that the core of NLS1 galaxies strongly tends toward the
red-side, with no significant difference between the cores of jetted and non-
jetted NLS1 galaxies.

  3. 3.

The redness of the core is most likely due to either dust extinction or due to
previous minor mergers.

With over 100 studied hosts of NLS1 galaxies, jetted and non-jetted, the host
galaxies of all types of NLS1 nuclei can be stated to be predominantly disk-
like. What is still mostly missing is the information regarding the host
galaxies of NLS1 sources at high redshifts, and more observations will be
needed to determine if their properties are similar to their low-redshift
counterparts. Suitable instruments for these studies would be, for example,
the Very Large Telescope as well as the JWST. Multi-wavelength observations
are crucial for understanding the role of mergers and interaction in jetted
NLS1 galaxies. For this, observations with the Atacama Large Millimeter Array
at sub-mm would be a key factor. NLS1 galaxies offer a unique opportunity to
study the interplay of different phenomena, such as jets, outflows, and star
formation, at an early stage of AGN evolution, and estimate the impact an
active nucleus has on the galaxy evolution.

###### Acknowledgements.

IRAF is distributed by the National Optical Astronomy Observatories, which are
operated by the Association of Universities for Research in Astronomy, Inc.,
under cooperative agreement with FourStar, an infrared camera, part of the
Magellan 6.5 m telescopes, at the Las Campanas Observatory in Chile. This
research has made use of GALFIT. This publication made use of the SIMBAD
database, operated at CDS, Strasbourg, France. This research has made use of
”Aladin sky atlas” developed at CDS, Strasbourg Observatory, France. This
research has made use of the NASA/IPAC Extragalactic Database (NED), which is
operated by the Jet Propulsion Laboratory, California Institute of Technology,
under contract with the National Aeronautics and Space Administration. This
research made use of Astropy,121212http://www.astropy.org a community-
developed core Python package for Astronomy (Astropy Collaboration et al.
2013, 2018). This publication makes use of data products from the Two Micron
All Sky Survey, which is a joint project of the University of Massachusetts
and the Infrared Processing and Analysis Center/California Institute of
Technology, funded by the National Aeronautics and Space Administration and
the National Science Foundation. IV would like to thank both the Magnus
Ehrnrooth Foundation and the Vilho, Yrjö and Kalle Väisälä Foundation of the
Finnish Academy of Science and Letters for their support. MB and EC are ESO
fellows.

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Therefore, the minimum of $\mathcal{I}({\sf P},{\sf Q})$ over ${\sf
Q}\in\mathcal{H}_{0}$ is obtained by ${\sf Q}_{*}$ such that for every
$e\in\mathcal{A}({\sf P})$

${\sf Q}_{*}^{e}\in\operatorname*{arg\,min}_{{\sf
Q}\in\mathcal{H}^{e}}\;\mathcal{I}\left({\sf P}^{e},{\sf
Q}\right)\quad\text{and}\quad{\sf Q}_{*}^{v_{0}}={\sf P}^{v_{0}},$

which yields

$\mathcal{I}({\sf P},\mathcal{H}_{0})=\mathcal{I}({\sf P},{\sf
Q}_{*})=\sum_{e\in\mathcal{A}({\sf P})}\mathcal{I}\left({\sf P}^{e},{\sf
Q}_{*}^{e}\right)+\mathcal{I}({\sf P},{\sf
Q}_{*};v_{0})=\sum_{e\in\mathcal{A}({\sf P})}\mathcal{I}\left({\sf
P}^{e},\mathcal{H}^{e}\right).$

∎

Now we are ready to prove Theorem 6.6.

###### Proof of Theorem 6.6.

First, fix any arbitrary ${\sf P}\in\mathcal{H}_{0}$. Then in order to prove
that this rule is asymptotically optimal under ${\sf P}$ as $\gamma,\delta\to
0$, it suffices to show that (48) holds when $s_{e}=e$ for every
$e\in\mathcal{K}$ since by Proposition D.1, this will further imply that (17)
in Theorem 4.4 is satisfied and the result will follow. Fix
$\bar{e}\in\mathcal{K}$ such that

$\bar{e}\in\operatorname*{arg\,min}_{e\in\mathcal{K}}\mathcal{I}\left({\sf
P}^{e},\mathcal{G}^{e}\right).$

Since $\mathcal{A}({\sf P})=\emptyset$ and (23) holds, $s$ is independent of
$s^{c}$, where $s=\bar{e}$. Furthermore, letting

${\sf Q}_{1}\in\operatorname*{arg\,min}_{{\sf
Q}\in\mathcal{G}_{\Psi,\bar{e}}}\mathcal{I}\left({\sf P},{\sf
Q};\bar{e}\right)\quad\text{implies that}\quad{\sf
Q}_{1}^{\bar{e}}\in\mathcal{G}^{\bar{e}}.$

This means under ${\sf Q}_{1}^{s}\otimes{\sf P}^{s^{c}}$ there is exactly one
dependent pair, i.e., $\mathcal{A}\left({\sf Q}_{1}^{s}\otimes{\sf
P}^{s^{c}}\right)=\\{\bar{e}\\}\in\Psi_{dis}$, which further implies

${\sf Q}_{1}^{s}\otimes{\sf
P}^{s^{c}}\in\mathcal{P}_{\Psi}\setminus\mathcal{H}_{0}.$

Thus, (48) holds.

Next, fix any arbitrary ${\sf
P}\in\mathcal{P}_{\Psi}\setminus\mathcal{H}_{0}$. We show that (47) holds when
$s^{\prime}_{e}=e$ for every $e\notin\mathcal{A}({\sf P})$ since by
Proposition D.1, this will further imply that (16) is satisfied. Fix the pair
$\bar{e}\notin\mathcal{A}({\sf P})$ such that

$\bar{e}\in\operatorname*{arg\,min}_{e\notin\mathcal{A}({\sf
P})}\;\mathcal{I}\left({\sf P}^{e},\mathcal{G}^{e}\right),$

and for which $s$ is independent of $s^{c}$, where
$s=s^{\prime}_{\bar{e}}=\bar{e}$. Furthermore, letting

${\sf Q}_{1}=\operatorname*{arg\,min}_{{\sf
Q}\in\mathcal{G}_{\Psi,\bar{e}}}\mathcal{I}\left({\sf P},{\sf
Q};\bar{e}\right)\quad\text{implies that}\quad{\sf
Q}_{1}^{\bar{e}}\in\mathcal{G}^{\bar{e}}.$

Since under ${\sf P}^{s^{c}}$ the dependent pairs among the sources $s^{c}$
are still disjoint and ${\sf Q}_{1}^{\bar{e}}\in\mathcal{G}^{\bar{e}}$, we
have $\bar{e}\in\mathcal{A}\left({\sf Q}_{1}^{s}\otimes{\sf
P}^{s^{c}}\right)\in\Psi_{dis}$. Thus,

${\sf Q}_{1}^{s}\otimes{\sf
P}^{s^{c}}\in\mathcal{P}_{\Psi}\setminus\mathcal{H}_{0},$

which implies (47) holds. Now we show that the conditions in Remark 4.4 are
satisfied.

First, we consider the case when $|\mathcal{A}({\sf P})|=1$. Let
$\mathcal{A}({\sf P})=\\{e_{1}\\}$. Since every ${\sf
Q}\in\cup_{e\in\mathcal{A}({\sf P})}\mathcal{H}_{\Psi,e}$ satisfies
$\mathcal{A}({\sf P})\setminus\mathcal{A}({\sf Q})\neq\emptyset$, we have
$e_{1}\notin\mathcal{A}({\sf Q})$, or ${\sf Q}^{e_{1}}\in\mathcal{H}^{e_{1}}$.
Thus, by Lemma B.2

$\mathcal{I}({\sf P},{\sf Q})\geq\mathcal{I}({\sf P}^{e_{1}},{\sf
Q}^{e_{1}})\geq\mathcal{I}({\sf P}^{e_{1}},\mathcal{H}^{e_{1}}),$

and further taking infimum over ${\sf Q}\in\cup_{e\in\mathcal{A}({\sf
P})}\mathcal{H}_{\Psi,e}$ in the left hand side yields

$\min_{e\in\mathcal{A}({\sf P})}\mathcal{I}({\sf
P},\mathcal{H}_{\Psi,e})\geq\mathcal{I}({\sf
P}^{e_{1}},\mathcal{H}^{e_{1}})=\mathcal{I}({\sf P},\mathcal{H}_{0}),$

where the equality follows from Lemma G.2. The equality also implies that (18)
in Theorem 4.5 is satisfied. Furthermore, since $s_{e}=s^{\prime}_{e}=e$ for
every $e\in\mathcal{K}$, we have

$\max_{e\in\mathcal{A}({\sf P})}\mathcal{I}({\sf
P},\mathcal{H}_{0};s_{e})=\mathcal{I}({\sf
P}^{e_{1}},\mathcal{H}^{e_{1}})=\min_{e\in\mathcal{A}({\sf
P})}\mathcal{I}({\sf P},\mathcal{H}_{\Psi,e};s^{\prime}_{e}).$

Thus, (19) holds.

Next, consider the case when $|\mathcal{A}({\sf P})|>1$. We will show that
(46) holds when $s^{\prime}_{e}=e$ for every $e\in\mathcal{A}({\sf P})$. Fix
$\bar{e}\in\mathcal{A}({\sf P})$ such that

$\bar{e}\in\operatorname*{arg\,min}_{e\in\mathcal{A}({\sf
P})}\;\mathcal{I}\left({\sf P}^{e},\mathcal{H}^{e}\right).$

Now since the dependent pairs are disjoint and (23) holds, it is clear that
$s$ is independent of $s^{c}$, where $s=\bar{e}$. Furthermore, letting

${\sf Q}_{1}=\operatorname*{arg\,min}_{{\sf
Q}\in\mathcal{H}_{\Psi,\bar{e}}}\mathcal{I}\left({\sf P},{\sf
Q};\bar{e}\right)\quad\text{implies that}\quad{\sf
Q}_{1}^{\bar{e}}\in\mathcal{H}^{\bar{e}}.$

Since under ${\sf Q}_{1}^{s}\otimes{\sf P}^{s^{c}}$, the dependent pairs are
still disjoint and $|\mathcal{A}\left({\sf Q}_{1}^{s}\otimes{\sf
P}^{s^{c}}\right)|=|\mathcal{A}({\sf P})|-1\geq 1$, we have

${\sf Q}_{1}^{s}\otimes{\sf
P}^{s^{c}}\in\mathcal{P}_{\Psi}\setminus\mathcal{H}_{0}.$

Thus, (46) holds and by Proposition D.1 this will further imply that (15) is
satisfied. This means

$\min_{e\in\mathcal{A}({\sf P})}\mathcal{I}({\sf
P},\mathcal{H}_{\Psi,e})=\min_{e\in\mathcal{A}({\sf P})}\mathcal{I}({\sf
P}^{e},\mathcal{H}^{e})\leq\sum_{e\in\mathcal{A}({\sf P})}\mathcal{I}({\sf
P}^{e},\mathcal{H}^{e})=\mathcal{I}({\sf P},\mathcal{H}_{0}),$

where the last equality follows from Lemma G.2. Furthermore, since
$s_{e}=s^{\prime}_{e}=e$ for every $e\in\mathcal{K}$, we have

$\max_{e\in\mathcal{A}({\sf P})}\mathcal{I}({\sf
P},\mathcal{H}_{0};s_{e})=\max_{e\in\mathcal{A}({\sf P})}\mathcal{I}({\sf
P}^{e},\mathcal{H}^{e})\geq\min_{e\in\mathcal{A}({\sf P})}\mathcal{I}({\sf
P}^{e},\mathcal{H}^{e})=\min_{e\in\mathcal{A}({\sf P})}\mathcal{I}({\sf
P},\mathcal{H}_{\Psi,e};s^{\prime}_{e}).$

Thus, (20) holds. Therefore, the asymptotic optimality of $\chi^{*}_{fwer}$
follows by using Theorem 4.6. The proof is complete. ∎

###  G.8 Proof of Corollary 6.1

###### Proof.

We have $\Psi=\Psi_{clus}$ and $s_{e}=s^{\prime}_{e}=e$ for every
$e\in\mathcal{K}$. Let $\mu$ be the mean vector and $\Sigma$ be the covariance
matrix of $X_{n}$ under ${\sf P}$. Then we have the decomposition that
$\Sigma=URU$, where $U$ is the diagonal matrix with elements of the main
diagonal being the standard deviations of the sources, i.e.,

$U={\rm{diag}}(\sigma_{1},\dots,\sigma_{K}),$

and $R$ is the correlation matrix. For any $k\in[K]$ we define

$A_{k}=(1-\rho)I_{k}+\rho 1_{k}1_{k}^{t},$

where $I_{k}$ is the $k\times k$ identity matrix, and $1_{k}$ is the column
vector in $\mathbb{R}^{k}$ with all elements being $1$. Furthermore, for any
$j,l\in[K]$ we denote by $0_{j\times l}$ the $j\times l$ matrix with all
entries being $0$. Without loss of generality let $v_{1}=\\{1,2,\dots,m\\}$,
then

$R=\begin{bmatrix}A_{m}&0_{m\times(K-m)}\\\ 0_{(K-m)\times
m}&I_{K-m}\end{bmatrix}.$

In the following we also use two important results regarding $A_{k}$:

$\displaystyle A_{k}^{-1}$
$\displaystyle=\frac{1}{1-\rho}I_{k}-\frac{\rho}{(1-\rho)(1+(k-1)\rho)}1_{k}1_{k}^{t},$
$\displaystyle det(A_{k})$ $\displaystyle=(1-\rho)^{k-1}(1+(k-1)\rho).$

  * •

Consider the distribution ${\sf Q}$ such that $L({\sf Q})=1$ and the data
sources $\\{1,\dots,m-1\\}$ marginally follow equicorrelated multivariate
Gaussian with correlation $\rho$ and is independent of the rest of the
sources. However, the mean vector of $X_{n}$ and the variances of the data
sources are the same under both ${\sf P}$ and ${\sf Q}$. Then, denoting by
$\Sigma_{{\sf Q}}$ the covariance matrix of $X_{n}$ under ${\sf Q}$, we have
$\Sigma_{\sf Q}=UR_{\sf Q}U$, where $R_{\sf Q}$ is the correlation matrix
under ${\sf Q}$ that takes the following form

$R_{\sf Q}=\begin{bmatrix}A_{m-1}&0_{(m-1)\times(K-m+1)}\\\
0_{(K-m+1)\times(m-1)}&I_{K-m+1}\end{bmatrix}.$

Then we have

$R_{\sf Q}^{-1}=\begin{bmatrix}B_{\sf Q}&0_{m\times(K-m)}\\\ 0_{(K-m)\times
m}&I_{K-m}\end{bmatrix},\quad\text{where}\quad B_{\sf
Q}=\begin{bmatrix}A_{m-1}^{-1}&0_{(m-1)\times 1}\\\
0_{1\times(m-1)}&1\end{bmatrix},$

which implies

$R_{\sf Q}^{-1}R=\begin{bmatrix}B_{\sf Q}A_{m}&0_{m\times(K-m)}\\\
0_{(K-m)\times m}&I_{K-m}\end{bmatrix}.$

Furthermore, we have

$\displaystyle B_{\sf Q}A_{m}=\begin{bmatrix}A_{m-1}^{-1}&0_{(m-1)\times 1}\\\
0_{1\times(m-1)}&1\end{bmatrix}\begin{bmatrix}A_{m-1}^{-1}&\rho 1_{m-1}\\\
\rho 1_{m-1}^{t}&1\end{bmatrix}=\begin{bmatrix}I_{m-1}&\rho
A_{m-1}^{-1}1_{m-1}\\\ \rho 1_{m-1}^{t}&1\end{bmatrix},$

which yields $tr(B_{\sf Q}A_{m})=m$. Now, from (14) we have

$\displaystyle\mathcal{I}({\sf P},{\sf Q})$
$\displaystyle=\frac{1}{2}\left\\{\log\frac{det(\Sigma_{\sf
Q})}{det(\Sigma)}-K+tr(\Sigma_{\sf Q}^{-1}\Sigma)+(\mu-\mu)^{t}\Sigma_{\sf
Q}^{-1}(\mu-\mu)\right\\}$
$\displaystyle=\frac{1}{2}\left\\{\log\frac{det(UR_{\sf
Q}U)}{det(URU)}-K+tr(U^{-1}R_{\sf Q}^{-1}U^{-1}URU)\right\\}$
$\displaystyle=\frac{1}{2}\left\\{\log\frac{(det(U))^{2}det(R_{\sf
Q})}{(det(U))^{2}det(R)}-K+tr(UU^{-1}R_{\sf Q}^{-1}R)\right\\}$
$\displaystyle=\frac{1}{2}\left\\{\log\frac{det(R_{\sf
Q})}{det(R)}-K+tr(R_{\sf Q}^{-1}R)\right\\}$
$\displaystyle=\frac{1}{2}\left\\{\log\frac{det(A_{m-1})}{det(A_{m})}-K+tr(B_{\sf
Q}A_{m})+tr(I_{K-m})\right\\}$
$\displaystyle=\frac{1}{2}\left\\{-\log\frac{det(A_{m})}{det(A_{m-1})}-K+m+(K-m)\right\\}$
$\displaystyle=-\frac{1}{2}\log\frac{(1-\rho)^{m-1}(1+(m-1)\rho)}{(1-\rho)^{m-2}(1+(m-2)\rho)}$
$\displaystyle=-\frac{1}{2}\log\frac{(1-\rho)(1+(m-1)\rho)}{(1+(m-2)\rho)}$
$\displaystyle=-\frac{1}{2}\log\left(1-\frac{\rho^{2}(m-1)}{1+(m-2)\rho}\right)$

Now note that, the pair $\\{m-1,m\\}$ is independent under ${\sf Q}$ but
dependent under ${\sf P}$, which implies ${\sf Q}\in\cup_{e\in\mathcal{A}({\sf
P})}\mathcal{H}_{\Psi,e}$. Therefore, as $\alpha,\beta,\gamma,\delta\to 0$
such that $|\log\gamma|>>|\log\alpha|\vee|\log\delta|$, from Theorem 6.2(ii)
we have

$\displaystyle{\sf{ARE}}_{{\sf P}}[\chi^{*}]$
$\displaystyle\leq\frac{\min_{e\in\mathcal{A}({\sf P})}\mathcal{I}\left({\sf
P},\mathcal{H}_{\Psi,e}\right)}{\min_{e\in\mathcal{A}({\sf
P})}\mathcal{I}\left({\sf
P}^{e},\mathcal{H}^{e}\right)}\leq\frac{\mathcal{I}({\sf P},{\sf
Q})}{(-1/2)\log(1-\rho^{2})}$
$\displaystyle=\log\left(1-\rho\frac{\rho(m-1)}{1+\rho(m-2)}\right)/\log(1-\rho^{2}).$

Since the above upper bound is an increasing function of $m$, one can let
$m\to\infty$ to obtain the following upper bound which is independent of $m$:

${\sf{ARE}}_{{\sf P}}[\chi^{*}]\leq\frac{\log(1-\rho)}{\log(1-\rho^{2})}.$

  * •

Consider the distribution ${\sf Q}$ such that $L({\sf Q})=1$ and the data
sources $\\{1,\dots,m+1\\}$ marginally follow equicorrelated multivariate
Gaussian with correlation $\rho$ and is independent of the rest of the
sources. However, the mean vector of $X_{n}$ and the variances of the data
sources are the same under both ${\sf P}$ and ${\sf Q}$. Then, denoting by
$\Sigma_{{\sf Q}}$ the covariance matrix of $X_{n}$ under ${\sf Q}$, we have
$\Sigma_{\sf Q}=UR_{\sf Q}U$, where $R_{\sf Q}$ is the correlation matrix
under ${\sf Q}$ that takes the following form

$R_{\sf Q}=\begin{bmatrix}A_{m+1}&0_{(m+1)\times(K-m-1)}\\\
0_{(K-m-1)\times(m+1)}&I_{K-m-1}\end{bmatrix}.$

Then we have

$R_{\sf Q}^{-1}=\begin{bmatrix}A_{m+1}^{-1}&0_{(m+1)\times(K-m-1)}\\\
0_{(K-m-1)\times(m+1)}&I_{K-m-1}\end{bmatrix},$

which implies

$R_{\sf Q}^{-1}R=\begin{bmatrix}A_{m+1}^{-1}B_{\sf
P}&0_{(m+1)\times(K-m-1)}\\\
0_{(K-m-1)\times(m+1)}&I_{K-m-1}\end{bmatrix},\quad\text{where}\quad B_{\sf
P}=\begin{bmatrix}A_{m}&0_{m\times 1}\\\ 0_{1\times m}&1\end{bmatrix}.$

Furthermore, we have

$\displaystyle A_{m+1}^{-1}B_{\sf P}$
$\displaystyle=\left(\frac{1}{1-\rho}I_{m+1}-\frac{\rho}{(1-\rho)(1+m\rho)}1_{m+1}1_{m+1}^{t}\right)B_{\sf
P}$ $\displaystyle=\frac{1}{1-\rho}B_{\sf
P}-\frac{\rho}{(1-\rho)(1+m\rho)}1_{m+1}1_{m+1}^{t}B_{\sf P},$

which yields

$\displaystyle tr(A_{m+1}^{-1}B_{\sf P})$
$\displaystyle=\frac{1}{1-\rho}tr(B_{\sf
P})-\frac{\rho}{(1-\rho)(1+m\rho)}tr(1_{m+1}1_{m+1}^{t}B_{\sf P})$
$\displaystyle=\frac{1}{1-\rho}(tr(A_{m})+1)-\frac{\rho}{(1-\rho)(1+m\rho)}tr(1_{m+1}^{t}B_{\sf
P}1_{m+1})$
$\displaystyle=\frac{m+1}{1-\rho}-\frac{\rho}{(1-\rho)(1+m\rho)}1_{m+1}^{t}B_{\sf
P}1_{m+1}$
$\displaystyle=\frac{m+1}{1-\rho}-\rho\frac{m(1+(m-1)\rho)+1}{(1-\rho)(1+m\rho)}$

Now, from (14) we have

$\displaystyle\mathcal{I}({\sf P},{\sf Q})$
$\displaystyle=\frac{1}{2}\left\\{\log\frac{det(\Sigma_{\sf
Q})}{det(\Sigma)}-K+tr(\Sigma_{\sf Q}^{-1}\Sigma)+(\mu-\mu)^{t}\Sigma_{\sf
Q}^{-1}(\mu-\mu)\right\\}$
$\displaystyle=\frac{1}{2}\left\\{\log\frac{det(UR_{\sf
Q}U)}{det(URU)}-K+tr(U^{-1}R_{\sf Q}^{-1}U^{-1}URU)\right\\}$
$\displaystyle=\frac{1}{2}\left\\{\log\frac{(det(U))^{2}det(R_{\sf
Q})}{(det(U))^{2}det(R)}-K+tr(UU^{-1}R_{\sf Q}^{-1}R)\right\\}$
$\displaystyle=\frac{1}{2}\left\\{\log\frac{det(R_{\sf
Q})}{det(R)}-K+tr(R_{\sf Q}^{-1}R)\right\\}$
$\displaystyle=\frac{1}{2}\left\\{\log\frac{det(A_{m+1})}{det(A_{m})}-K+tr(A_{m+1}^{-1}B_{\sf
P})+tr(I_{K-m-1})\right\\}$
$\displaystyle=\frac{1}{2}\bigg{\\{}\log\frac{(1-\rho)^{m}(1+m\rho)}{(1-\rho)^{m-1}(1+(m-1)\rho)}-K+K-m-1$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+\frac{m+1}{1-\rho}-\rho\frac{m(1+(m-1)\rho)+1}{(1-\rho)(1+m\rho)}\bigg{\\}}$
$\displaystyle=\frac{1}{2}\bigg{\\{}\log\frac{(1-\rho)(1+m\rho)}{(1+(m-1)\rho)}$
$\displaystyle+\frac{(m+1)(1+m\rho)-\rho(1+m+m(m-1)\rho)-(m+1)(1-\rho)(1+m\rho)}{(1-\rho)(1+m\rho)}\bigg{\\}}$
$\displaystyle=\frac{1}{2}\bigg{\\{}-\log\frac{(1+(m-1)\rho)}{(1-\rho)(1+m\rho)}+\rho\frac{(m+1)(1+m\rho)-(1+m+m(m-1)\rho)}{(1-\rho)(1+m\rho)}\bigg{\\}}$
$\displaystyle=-\frac{1}{2}\bigg{\\{}-\log\left(\frac{1+\rho(m-1)}{1+\rho(m-1)-\rho^{2}m}\right)+\frac{2m\rho^{2}}{(1-\rho)(1+m\rho)}\bigg{\\}}$
$\displaystyle=-\frac{1}{2}\log\left(\frac{1+\rho(m-1)}{1+\rho(m-1)-\rho^{2}m}\right)+\frac{m\rho^{2}}{(1+m\rho)(1-\rho)}$

Now note that, the pair $\\{m,m+1\\}$ is dependent under ${\sf Q}$ but
independent under ${\sf P}$, which implies ${\sf
Q}\in\cup_{e\notin\mathcal{A}({\sf P})}\mathcal{G}_{\Psi,e}$. Therefore, as
$\alpha,\beta,\gamma,\delta\to 0$ such that
$|\log\delta|>>|\log\alpha|\vee|\log\gamma|$, from Theorem 6.3(ii) we have

$\displaystyle{\sf{ARE}}_{{\sf
P}}[\chi^{*}]\leq\frac{\min_{e\notin\mathcal{A}({\sf P})}\mathcal{I}\left({\sf
P},\mathcal{G}_{\Psi,e}\right)}{\min_{e\notin\mathcal{A}({\sf
P})}\mathcal{I}\left({\sf
P}^{e},\mathcal{G}^{e}\right)}\leq\frac{\mathcal{I}({\sf P},{\sf
Q})}{\log\sqrt{1-\rho^{2}}+{\rho^{2}}/{(1-\rho^{2})}}$
$\displaystyle\leq\left\\{-\frac{1}{2}\log\left(\frac{1+\rho(m-1)}{1+\rho(m-1)-\rho^{2}m}\right)+\frac{m\rho^{2}}{(1+m\rho)(1-\rho)}\right\\}\bigg{/}\left\\{\log\sqrt{1-\rho^{2}}+\frac{\rho^{2}}{1-\rho^{2}}\right\\}.$

Since the above upper bound is an increasing function of $m$, one can let
$m\to\infty$ to obtain the following upper bound which is independent of $m$:

${\sf{ARE}}_{{\sf
P}}[\chi^{*}]\leq\left(\log(1-\rho)+\frac{2\rho}{1-\rho}\right)\bigg{/}\left(\log(1-\rho^{2})+\frac{2\rho^{2}}{1-\rho^{2}}\right).$

  * •

Consider the distribution ${\sf Q}$ such that all sources are independent.
However, the mean vector of $X_{n}$ and the variances of the data sources are
the same under both ${\sf P}$ and ${\sf Q}$. Then, denoting by $\Sigma_{{\sf
Q}}$ the covariance matrix of $X_{n}$ under ${\sf Q}$, we have $\Sigma_{\sf
Q}=U^{2}$. Now, from (14) we have

$\displaystyle\mathcal{I}({\sf P},{\sf Q})$
$\displaystyle=\frac{1}{2}\left\\{\log\frac{det(\Sigma_{\sf
Q})}{det(\Sigma)}-K+tr(\Sigma_{\sf Q}^{-1}\Sigma)+(\mu-\mu)^{t}\Sigma_{\sf
Q}^{-1}(\mu-\mu)\right\\}$
$\displaystyle=\frac{1}{2}\left\\{\log\frac{det(U^{2})}{det(URU)}-K+tr(U^{-1}U^{-1}URU)\right\\}$
$\displaystyle=\frac{1}{2}\left\\{\log\frac{(det(U))^{2}}{(det(U))^{2}det(R)}-K+tr(UU^{-1}R)\right\\}$
$\displaystyle=\frac{1}{2}\left\\{\log\frac{1}{det(R)}-K+tr(R)\right\\}$
$\displaystyle=-\frac{1}{2}\log(det(R))=-\frac{1}{2}\log(det(A_{m}))$
$\displaystyle=-\frac{1}{2}\log((1-\rho)^{m-1}(1+(m-1)\rho))$
$\displaystyle=-\frac{1}{2}\left((m-1)\log(1-\rho)+\log(1+(m-1)\rho)\right)$

Now, clearly ${\sf Q}\in\mathcal{H}_{0}$. Therefore, as
$\alpha,\beta,\gamma,\delta\to 0$ such that
$|\log\alpha|>>|\log\gamma|\vee|\log\delta|$, from Theorem 6.4(ii) we have

$\displaystyle{\sf{ARE}}_{{\sf P}}[\chi^{*}]$
$\displaystyle\leq\min_{e\in\mathcal{A}({\sf
P})}\left\\{\frac{\mathcal{I}({\sf P},\mathcal{H}_{0})}{\mathcal{I}({\sf
P}^{e},\mathcal{H}^{e})}\right\\}\leq\frac{\mathcal{I}({\sf P},{\sf
Q})}{(-1/2)\log(1-\rho^{2})}$
$\displaystyle=\frac{(m-1)\log(1-\rho)+\log(1+(m-1)\rho)}{\log(1-\rho^{2})}.$

The proof is complete. ∎

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