# Simultaneous Millimeter-wave, Gamma-ray, and Optical Monitoring of the
Blazar PKS 2326-502 During a Flaring State

J. C. Hood II Kavli Institute for Cosmological Physics, University of Chicago,
5640 South Ellis Avenue, Chicago, IL, USA 60637  Department of Astronomy and
Astrophysics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL, USA
60637  Department of Physics and Astronomy, Vanderbilt University, Nashville,
TN, USA 37235  A. Simpson  Department of Physics, Rensselaer Polytechnic
Institute, Troy, NY 12180  A. McDaniel Department of Physics and Astronomy,
Clemson University, Clemson, SC, 29631  A. Foster Department of Physics, Case
Western Reserve University, Cleveland, OH, 44106, USA  P. A. R. Ade  Cardiff
University, Cardiff CF10 3XQ, United Kingdom  M. Ajello  Department of Physics
and Astronomy, Clemson University, Clemson, SC, 29631  A. J. Anderson  Fermi
National Accelerator Laboratory, MS209, P.O. Box 500, Batavia, IL 60510  J. E.
Austermann  NIST Quantum Devices Group, 325 Broadway Mailcode 817.03, Boulder,
CO, USA 80305  Department of Physics, University of Colorado, Boulder, CO, USA
80309  J. A. Beall  NIST Quantum Devices Group, 325 Broadway Mailcode 817.03,
Boulder, CO, USA 80305  A. N. Bender  High Energy Physics Division, Argonne
National Laboratory, 9700 S. Cass Avenue, Argonne, IL, USA 60439  Kavli
Institute for Cosmological Physics, University of Chicago, 5640 South Ellis
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Laboratory, MS209, P.O. Box 500, Batavia, IL 60510  Kavli Institute for
Cosmological Physics, University of Chicago, 5640 South Ellis Avenue, Chicago,
IL, USA 60637  Department of Astronomy and Astrophysics, University of
Chicago, 5640 South Ellis Avenue, Chicago, IL, USA 60637  F. Bianchini School
of Physics, University of Melbourne, Parkville, VIC 3010, Australia  L. E.
Bleem High Energy Physics Division, Argonne National Laboratory, 9700 S. Cass
Avenue, Argonne, IL, USA 60439  Kavli Institute for Cosmological Physics,
University of Chicago, 5640 South Ellis Avenue, Chicago, IL, USA 60637  J. E.
Carlstrom Kavli Institute for Cosmological Physics, University of Chicago,
5640 South Ellis Avenue, Chicago, IL, USA 60637  Department of Physics,
University of Chicago, 5640 South Ellis Avenue, Chicago, IL, USA 60637  High
Energy Physics Division, Argonne National Laboratory, 9700 S. Cass Avenue,
Argonne, IL, USA 60439  Department of Astronomy and Astrophysics, University
of Chicago, 5640 South Ellis Avenue, Chicago, IL, USA 60637  Enrico Fermi
Institute, University of Chicago, 5640 South Ellis Avenue, Chicago, IL, USA
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60439  Department of Astronomy and Astrophysics, University of Chicago, 5640
South Ellis Avenue, Chicago, IL, USA 60637  P. Chaubal  School of Physics,
University of Melbourne, Parkville, VIC 3010, Australia  H. C. Chiang
Department of Physics, McGill University, 3600 Rue University, Montreal,
Quebec H3A 2T8, Canada  School of Mathematics, Statistics & Computer Science,
University of KwaZulu-Natal, Durban, South Africa  T-L. Chou  Kavli Institute
for Cosmological Physics, University of Chicago, 5640 South Ellis Avenue,
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Kavli Institute for Cosmological Physics, University of Chicago, 5640 South
Ellis Avenue, Chicago, IL, USA 60637  Department of Astronomy and
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Chicago, 5640 South Ellis Avenue, Chicago, IL, USA 60637  Department of
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University of Toronto, 50 St George St, Toronto, ON, M5S 3H4, Canada
Department of Astronomy & Astrophysics, University of Toronto, 50 St George
St, Toronto, ON, M5S 3H4, Canada  T. de Haan  Institute of Particle and
Nuclear Studies (IPNS), High Energy Accelerator Research Organization (KEK),
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Japan  M. A. Dobbs  Department of Physics, McGill University, 3600 Rue
University, Montreal, Quebec H3A 2T8, Canada  Canadian Institute for Advanced
Research, CIFAR Program in Gravity and the Extreme Universe, Toronto, ON, M5G
1Z8, Canada  W. Everett  Department of Astrophysical and Planetary Sciences,
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Urbana, IL 61801, USA  Canadian Institute for Advanced Research, CIFAR Program
in Gravity and the Extreme Universe, Toronto, ON, M5G 1Z8, Canada  W. L.
Holzapfel  Department of Physics, University of California, Berkeley, CA, USA
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and Cosmology, SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA
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Angeles, CA 90095-1547, USA  J. J. McMahon  Kavli Institute for Cosmological
Physics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL, USA
60637  Department of Physics, University of Chicago, 5640 South Ellis Avenue,
Chicago, IL, USA 60637  Department of Astronomy and Astrophysics, University
of Chicago, 5640 South Ellis Avenue, Chicago, IL, USA 60637  S. S. Meyer
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Ellis Avenue, Chicago, IL, USA 60637  Department of Physics, University of
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South Ellis Avenue, Chicago, IL, USA 60637  J. Montgomery  Department of
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Canada  T. Natoli  Department of Astronomy and Astrophysics, University of
Chicago, 5640 South Ellis Avenue, Chicago, IL, USA 60637  Kavli Institute for
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IL, USA 60637  J. P. Nibarger  NIST Quantum Devices Group, 325 Broadway
Mailcode 817.03, Boulder, CO, USA 80305  G. Noble  Department of Physics,
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Cass Avenue, Argonne, IL, USA 60439  Y. Omori  Dept. of Physics, Stanford
University, 382 Via Pueblo Mall, Stanford, CA 94305  S. Padin  Kavli Institute
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of Chicago, 5640 South Ellis Avenue, Chicago, IL, USA 60637  California
Institute of Technology, 1200 E. California Blvd., Pasadena CA 91125 USA  S.
Patil  School of Physics, University of Melbourne, Parkville, VIC 3010,
Australia  C. Pryke  School of Physics and Astronomy, University of Minnesota,
116 Church Street S.E. Minneapolis, MN, USA 55455  C. L. Reichardt School of
Physics, University of Melbourne, Parkville, VIC 3010, Australia  J. E. Ruhl
Department of Physics, Case Western Reserve University, Cleveland, OH, 44106,
USA  Center for Education and Research in Cosmology and Astrophysics, Case
Western Reserve University, Cleveland, OH, USA 44106  B. R. Saliwanchik
Physics Department, Center for Education and Research in Cosmology and
Astrophysics, Case Western Reserve University, Cleveland, OH, USA 44106
Department of Physics, Yale University, P.O. Box 208120, New Haven, CT
06520-8120  K. K. Schaffer  Kavli Institute for Cosmological Physics,
University of Chicago, 5640 South Ellis Avenue, Chicago, IL, USA 60637  Enrico
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USA 60637  Liberal Arts Department, School of the Art Institute of Chicago,
112 S Michigan Ave, Chicago, IL, USA 60603  C. Sievers  University of Chicago,
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Physics, McGill University, 3600 Rue University, Montreal, Quebec H3A 2T8,
Canada  Three-Speed Logic, Inc., Victoria, B.C., V8S 3Z5, Canada  A. A. Stark
Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA,
USA 02138  C. Tucker  Cardiff University, Cardiff CF10 3XQ, United Kingdom  T.
Veach  Space Science and Engineering Division, Southwest Research Institute,
San Antonio, TX 78238  J. D. Vieira  Astronomy Department, University of
Illinois at Urbana-Champaign, 1002 W. Green Street, Urbana, IL 61801, USA
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Street, Urbana, IL 61801, USA  Center for AstroPhysical Surveys, National
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Wu Kavli Institute for Cosmological Physics, University of Chicago, 5640 South
Ellis Avenue, Chicago, IL, USA 60637  SLAC National Accelerator Laboratory,
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Division, Argonne National Laboratory, 9700 S. Cass Avenue, Argonne, IL, USA
60439  J. A. Zebrowski  Department of Physics, University of California,
Berkeley, CA, USA 94720  L. Zhang Department of Physics, University of
California, Santa Barbara, CA 93106, USA  Department of Physics, University of
Illinois at Urbana-Champaign, 1110 West Green St., Urbana, IL 61801, USA
Department of Astronomy, University of Illinois at Urbana-Champaign, 1002 West
Green St., Urbana, IL 61801, USA

###### Abstract

Including millimeter-wave (mm-wave) data in multi-wavelength studies of the
variability of active galactic nuclei (AGN) can provide insights into AGN
physics that are not easily accessible at other wavelengths. We demonstrate in
this work the potential of cosmic microwave background (CMB) telescopes to
provide long-term, high-cadence mm-wave AGN monitoring over large fractions of
sky. We report on a pilot study using data from the SPTpol instrument on the
South Pole Telescope (SPT), which was designed to observe the CMB at arcminute
and larger angular scales. Between 2013 and 2016, SPTpol was used primarily to
observe a single 500 deg2 field, covering the entire field several times per
day with detectors sensitive to radiation in bands centered at 95 and 150 GHz.
We use SPT 150 GHz observations to create AGN light curves, and we compare
these mm-wave light curves to those at other wavelengths, in particular
$\gamma$-ray and optical. In this Letter, we focus on a single source, PKS
2326-502, which has extensive, day-timescale monitoring data in gamma-ray,
optical, and now mm-wave between 2013 and 2016. We find PKS 2326-502 to be in
a flaring state in the first two years of this monitoring, and we present a
search for evidence of correlated variability between mm-wave, optical R band,
and $\gamma$-ray observations. This pilot study is paving the way for AGN
monitoring with current and upcoming CMB experiments such as SPT-3G, Simons
Observatory, and CMB-S4, including multi-wavelength studies with facilities
such as VRO-LSST.

AGN — Gamma-ray source — Millimeter astronomy

††facilities: SPT, FERMI, SMARTS

##  1 Introduction

Active galactic nuclei (AGN) are accreting supermassive ($M\gtrsim
10^{5}M_{\odot}$) black holes commonly found at the centers of massive
galaxies (e.g., Kormendy & Richstone, 1995; Gebhardt et al., 2000). The
Unified Model of AGN proposes to explain observed categories of AGN via a
scenario in which the appearance of a source depends on the angle between the
axis of symmetry of the source and the line of sight of the observer (e.g.,
Antonucci, 1993; Urry & Padovani, 1995). For example, in this scenario,
blazars—radio-loud AGN111Radio-loud AGN are generally defined as AGN with a
ratio of radio (5 GHz) to optical (B-band) flux $\geq$ 10 (Kellermann et al.,
1989). that also emit strongly in the $\gamma$-ray band—are understood to have
a relativistic jet pointed at relatively small angles ($<$5 deg) to the
observer. The spectral energy distribution (SED) of blazars has a
characteristic double-humped structure, with one peak located anywhere from
the high-frequency radio to the soft X-ray band, caused by synchrotron
emission from energetic electrons in the blazar jet, and a high-energy peak in
the MeV-TeV $\gamma$-ray band (e.g., Fossati et al., 1998).

The source of the high-energy peak is still under debate, with models for the
production of $\gamma$-ray photons classified into two broad classes: hadronic
and leptonic models (e.g., Blandford et al., 2019). In hadronic models,
processes such as photo-pion production are responsible for the $\gamma$-ray
peak, while in leptonic models, the $\gamma$-ray peak is caused by inverse-
Compton scattering of lower-energy photons, which can be the same synchrotron
photons responsible for the low-energy peak (the “synchrotron self-Compton”
model) or other components of the radiation field (the “external inverse-
Compton” model, e.g., Sikora et al. 1994).

A key to distinguishing between these models is what they predict for multi-
wavelength observations of blazar flares. Leptonic models have been successful
in explaining several observed aspects of blazars (Sikora et al., 1994; Sikora
& Madejski, 2003). The simplest interpretation of leptonic models predict that
when observing AGN light curves in multiple wavelengths, there should be
correlated variability between the synchrotron peak and the high-energy peak.
This behavior has been observed in many cases (e.g., Bonning et al. 2009), but
evidence exists that it may not always be present. For example, in the multi-
wavelength study of PKS 0208-502, an “orphan flare” was observed, in which a
significant flux increase is seen in the optical/infrared bands but not in the
$\gamma$-ray band (Chatterjee et al., 2013a, b).

Multi-wavelength studies of blazar flares have traditionally included
$\gamma$-ray, X-ray, optical, infrared, and radio emission. Since millimeter-
wavelength (mm-wave) radiation is a strong tracer of synchrotron emission,
observations of AGN at these wavelengths should help identify the true origin
of the blazar SED. Recent studies have shown that on longer timescales, mm-
wave variability is better correlated with $\gamma$-ray emission than optical
(Meyer et al., 2019; Zhang et al., 2022), while on shorter timescales features
tend to correlate more between the optical and $\gamma$-ray. This points
toward the possibility of synchrotron emission produced in different regions
of the blazar being responsible for the mm–$\gamma$-ray correlation and the
optical–$\gamma$-ray correlation.

It has recently been recognized that cosmic microwave background (CMB)
experiments have the potential to be used as AGN monitors (e.g., Holder et
al., 2019). AGN appear as bright point sources in maps made with CMB
experiments, and current CMB experiments are sufficiently sensitive to detect
many AGN at high signal-to-noise ratio (S/N) in short observations. When
combined with an observing strategy that results in high-cadence observations
of the same patch of sky over many years, CMB data-sets are effective for AGN
monitoring.

We have undertaken a pilot study of AGN variability using mm-wave data from
SPTpol, the second-generation camera on the South Pole Telescope (SPT). The
SPTpol survey enables the monitoring of tens of mm-bright AGN on timescales
from days to years at high S/N ($>$ 10 in a 36-hour coadd). These observations
provide the opportunity to include high-cadence mm-wave data in the study of
the physical mechanisms behind AGN emission.

Although our SPTpol AGN monitoring campaign includes tens of sources, we
choose to focus on the blazar PKS 2326-502 for this pilot study because of its
long history of observations in multiple wavelengths (e.g., Dutka et al.,
2017). PKS 2326-502 is among the targets of monitoring by both the Fermi Large
Area Telescope (LAT) and Yale Small and Moderate Aperture Research Telescope
System (SMARTS) Blazar Group collaborations. In particular, PKS 2326-502 has
publicly available Fermi ($\gamma$-ray) and SMARTS (optical) observations over
most of the time period over which we have SPTpol data.

##  2 Observations

In our study of PKS 2326-502, we use data from SPT, SMARTS, and Fermi-LAT. In
this section, we describe the observations and data reduction for each
instrument.

###  2.1 SPT

The SPT (Carlstrom et al., 2011) is a 10-meter telescope located at the
geographic South Pole and dedicated to making low-noise, high-resolution maps
of the mm-wave sky, with the primary goal of mapping the temperature and
polarization anisotropies in the CMB. Three separate cameras have been
installed on the telescope, each used to map multiple large patches of the
Southern Celestial Hemisphere. This work uses data from the second-generation
camera, SPTpol. From 2013 to 2016 SPTpol was used during most of the year to
survey 500 $\mathrm{deg^{2}}$ of the southern extragalactic sky at arcminute
resolution to mJy noise levels in bands centered at 95 and 150 GHz. The 500
$\mathrm{deg^{2}}$ SPTpol survey consists of $\sim$3500 observations of a
field covering $22^{h}$ to $2^{h}$ in right ascension and -$65^{\circ}$ to
-$50^{\circ}$ in declination (Henning et al., 2018). For this study we take
150 GHz maps made from individual observations and combine them into 36-hour
bundles, which provides a reasonable match with the cadence of other datasets
while also providing high S/N on a sufficient number of sources.

Once bundle maps are created, we apply a matched filter that removes the long-
wavelength modes from each map, maximizing the S/N on point sources. These
filtered bundles have a 1$\sigma$ error of $\sim$9 mJy, providing us with S/N
$>10$ on 25 AGN in the 500 $\mathrm{deg^{2}}$ field. We perform a series of
calibration and systematic checks for each bundle. We check and correct per-
bundle astrometry by comparing the positions of bright sources to those in the
AT20G catalog (Murphy et al., 2010). We correct the calibration of each bundle
by calculating the cross-spectrum of that bundle map with the Planck 143 GHz
map (Planck Collaboration et al., 2020) and scaling the bundle map so that the
cross-spectrum agrees with that calculated for the average of all bundles. We
additionally check for contamination in each bundle (such as from sidelobe
pickup from the Sun in observations during the Austral summer, which would
appear as bright streaks in our observations) by visually inspecting
$5^{\prime}\times 5^{\prime}$ patches of sky centered on PKS 2326-502. No such
contamination was detected. Once all maps have been calibrated and checked, we
extract the fluxes that are used to create the mm-wave light curve of PKS
2326-502.

###  2.2 SMARTS

This paper makes use of optical/near-infrared light curves that are available
at the SMARTS website.222www.astro.yale.edu/smarts/glast/home.php The SMARTS
telescope is located in Cerro Tololo Chile, and is thus well suited to
monitoring of Southern Hemisphere targets. The SMARTS blazar sample was
initially (in 2008) defined to include all Fermi-LAT-monitored blazars on the
initial public release list with declination $<$ $20^{\circ}$. Observations
were made in the B, V, R, J & K bands, with an observing cadence of 1 to 3
days. Here, we use the 1-day cadence optical R band observations to match the
SPT cadence as closely as possible. The full details of the data selection and
analysis procedure for SMARTS data can be found in Bonning et al. (2012).

###  2.3 Fermi LAT

The Fermi-LAT light curve for PKS 2326-502 is taken from the Fermi-LAT Light
Curve Repository (LCR, Abdollahi et al.
2023).333https://fermi.gsfc.nasa.gov/ssc/data/access/lat/LightCurveRepository
The LCR is a public database of multi-cadence flux-calibrated light curves for
over 1500 variable sources in the 10-year Fermi-LAT point source catalog
(4FGL-DR2, Ballet et al. 2020). The light curves generated by the LCR span the
duration of the mission and are obtained by performing an unbinned likelihood
analysis over the energy range $100\,\mathrm{MeV}–100$ GeV. The LCR analysis
uses the standard Fermi-LAT Science Tools (version v11r5p3) and the
P8R2_SOURCE_V6 instrument response functions on P8R3_SOURCE class photons.
Photons are selected from a $12^{\circ}$ region of interest (ROI) centered on
the location of the 4FGL-DR2 counterpart of PKS 2326-502 (4FGL J2329.3-4955).
A zenith angle cut of $90^{\circ}$ is used to prevent contamination from the
Earth’s limb. Included in the photon distribution model used to calculate the
flux of the target source are all 4FGL-DR2 point sources within $30^{\circ}$
as well as Galactic diffuse (gll_iem_v07.fits) and isotropic
(iso_P8R3_SOURCE_V3_v1) background models. The LCR provides light curves in
cadences of 3 days, 1 week, and 1 month. For this analysis we use the minimum
available time binning of 3 days.

##  3 Methods

In this Letter, we report both qualitative and quantitative results from the
analysis of multi-wavelength light curves of PKS 2326-502. Quantitatively, we
measure the local cross-correlation functions (CCFs)444Here “local” refers to
calculating the mean and variance of both light curves over individual time
lag bins rather than the entire light curve, see Welsh (1999) for details. of
year-long light curves and calculate the significance by comparing these to
uncorrelated simulations. The simulations were created by taking the power
spectrum of the light curve from each data set, fitting to a model in which
the light curve fluctuation power as a function of temporal frequency
$P(f)=P_{0}(1+(f/f_{\mathrm{knee}})^{-\alpha})$, and producing 10,000
simulations of light curves from each model power spectrum. The simulated
light curves are generated in Fourier space with random phase (i.e., they obey
Gaussian statistics in real space). Some recent results have indicated that,
at least in the $\gamma$-ray, blazar variability is better described by a log-
normal probability distribution than a Gaussian (e.g., Duda & Bhatta, 2021).
We have created an alternate set of simulations with log-normal statistics and
do not see any significant change in our results when we use this alternate
set.

We calculate the local CCF for each pair of light curves in the real data and
all 10,000 simulations using the following procedure. For a given time lag bin
$\tau$, we select all data points in light curves $a$ and $b$ that satisfy:

$t(a)-t(b)\in\tau\pm\frac{\Delta\tau}{2},$ (1)

where $\Delta\tau$ is the bin width, and $t(a)$ and $t(b)$ are the times for
observations in each light curve. We define the local CCF as:

$\mathrm{CCF}(\tau)=\frac{\sum_{i=1}^{n}(a_{i}-\bar{a})(b_{i}-\bar{b})}{(n-1)s_{a}s_{b}},$
(2)

where $s_{a}$ is defined as:

$s_{a}=\sqrt{\frac{1}{n-1}\sum_{i=1}^{n}(a_{i}-\bar{a})^{2}},$ (3)

$\bar{a}$ is defined as:

$\bar{a}=\frac{1}{n}\sum_{i=1}^{n}a_{i}$ (4)

and $i$ runs over all pairs of points that satisfy Equation 1. The simulated
CCFs were then used to find the 1$\sigma$, 2$\sigma$ and 3$\sigma$ contours
for the data CCFs.

As discussed in Welsh (1999), measuring the full correlation function is
challenging in data that is dominated by the longest time-scale feature in the
data. We also wish to remove possible dependencies on the binning timescale of
any of our data sets. For these reasons, we boxcar-smooth all of the light
curves with a 7-day window, and we only calcluate the full CCF on data that
has been detrended using a polynomial filter. We use a fifth-order polynomial
per year, which preserves features up to time scales of months. For data that
has not been detrended, we only calculate the zero-lag correlation and
associated p-value. This p-value is estimated by calculating the number of
simulations that have a higher zero-lag correlation than the data:

$p(\mathrm{CCF},\tau=0)=\frac{N(\mathrm{CCF}_{\mathrm{sim}},\tau=0>\mathrm{CCF}_{\mathrm{data}},\tau=0)}{N_{\mathrm{sims}}}$
(5)

For detrended data, we calculate this zero-lag correlation and p-value, and we
further plot the full CCF and look for evidence of lags between the flaring in
different bands.

##  4 Results

Multi-wavelength ($\gamma$-ray, optical, mm-wave) light curves for four years
of monitoring of PKS 2326-502 are shown in Figure 1. We note that the raw
statistical significance of the variability in all three bands is high: The
typical S/N in a single 36-hour SPT light curve point is $\sim$50 in the
quiescent state and over 200 in the flaring state, and the corresponding S/N
for the 3-day Fermi light curve points are 1–2 and 7–10. For SMARTS-R, where
we only have data in the flaring state, the typical S/N per 1-day point is
$\sim$50.

Several features of these light curves which make up the primary results of
this Letter are evident by-eye in Figure 1, including:

Figure 1: Light curves for PKS 2326-502. Top: Fermi-LAT; Middle: SMARTS
optical R; Bottom: SPTpol 150GHz. Evident by eye are long-timescale
correlation between mm-wave and $\gamma$-ray observations and short-timescale
correlation between optical and $\gamma$-ray observations. For reference, MJD
56400 was calendar date April, 18 2013. The time gaps in the SPTpol data are
periods during the austral summer when the primary 500 $\mathrm{deg^{2}}$
field was not observed to avoid solar contamination.

  1. 1.

A long-timescale flaring state in the first two years followed by a two-year
quiescent period.

  2. 2.

Long-timescale correlation between mm-wave and $\gamma$-ray data, with the mm-
wave light curve appearing to decay more slowly than the $\gamma$-ray one.

  3. 3.

Short-timescale correlation between $\gamma$-ray and optical data.

For our quantitative analyses, we focus on the observations made in the first
two years of available SPTpol data (2013-2014), because: 1) PKS 2326-502
entered into a quiescent state thereafter, and 2) there is no publicly
available optical data from SMARTS after 2014. For all possible pairs of data,
two sets of light curves (boxcar-smoothed and smoothed-and-detrended) and CCFs
for the smoothed and detrended data are shown for year 1 and year 2 in Figure
2 and Figure 3 respectively.

As a rough measure of the significance of the correlated year time-scale flare
in the $\gamma$-ray and mm-wave bands, we calculate the number of simulations
that show a similar or larger flux increase over one year in those two bands.
We find that only 42 out of 10,000 simulations show a factor of 2.5 increase
over one year in both bands. We chose a factor of 2.5 because the ratio of the
flux in the first and last month of year one was 2.7 in SPT and 3.3 in Fermi.
Therefore, we report $4.2\times 10^{-3}$ as a raw, non-trials-corrected
p-value for this long-timescale correlated flaring state. We also calculate
the zero-lag correlation for the unfiltered boxcar-smoothed year one data and
we find a zero-lag correlation value of 0.75 for SPT x Fermi. Only 287
simulations show a zero-lag correlation between SPT x Fermi higher than this,
thus we report in Table 1 a p-value of 0.03 for this correlation.

Another fairly strong identifiable feature in the data is the short
$\sim$week-timescale flare seen in both Fermi and SMARTS but not in SPT. This
leads to a significant detection of zero-lag correlation even in the non-
detrended data $p(\mathrm{CCF},\tau=0)=3\times 10^{-4}$. Once we filter out
the long time-scale features we find a zero-lag correlation value much higher
than found in any of our simulations and thus report a p-value of $<10^{-4}$
in Table 1. This confirms that this correlation is being driven by the
shorter-timescale feature in the Fermi and SMARTS light curves shown in Figure
2. By contrast, when we detrend SPT x Fermi data in year one, we find no
evidence of correlation on shorter timescales.

Figure 2: Year one boxcar-smoothed light curves (left column), smoothed and
detrended light curves (middle column), and detrended CCFs (right column). The
grey shaded regions represent the 1$\sigma$, 2$\sigma$ and 3$\sigma$ contours
of simulated CCFs.

In contrast to year one, for year two we find a significant correlation
between the detrended SPT x Fermi light curves, but none for the SMARTS x
Fermi light curves, as shown in Figure 3. We also find no significant
correlation between any of the data sets in year two prior to detrending.
Finally we note that we measure no significant correlation at non-zero lag for
any data combination in either year.

Figure 3: Same as Figure 2 but for year 2. Light Curve Statistics
---
Dataset | zero-lag correlation | zero-lag p-value
SPT x Fermi year one (smoothed) | 0.75 | $2.9\times 10^{-2}$
SPT x Fermi year one (smoothed & detrended) | $-1.3\times 10^{-2}$ | 0.52
Smarts x Fermi year one (smoothed) | 0.92 | $3.0\times 10^{-4}$
Smarts x Fermi year one (smoothed & detrended) | 0.92 | $<10^{-4}$
SPT x Smarts year one (smoothed) | 0.48 | 0.16
SPT x Smarts year one (smoothed & detrended) | $2.6\times 10^{-2}$ | 0.47
SPT x Fermi year two (smoothed) | 0.23 | 0.32
SPT x Fermi year two (smoothed & detrended) | 0.67 | $1.0\times 10^{-3}$
Smarts x Fermi year two (smoothed) | 0.54 | $7.4\times 10^{-2}$
Smarts x Fermi year two (smoothed & detrended) | 0.34 | $8.7\times 10^{-2}$
SPT x Smarts year two (smoothed) | 0.32 | 0.26
SPT x Smarts year two (smoothed & detrended) | $9.7\times 10^{2}$ | 0.36
Table 1: Values of zero-lag correlation and associated p-value for smoothed
and smoothed-and-detrended data. Values for years one and two are reported
separately.

##  5 Discussion

Our study of the multi-wavelength variability of PKS 2326-502 yields four
primary results:

  1. 1.

Long-timescale correlation between mm-wave and $\gamma$-ray data, with the mm-
wave light curve appearing to decay more slowly than the $\gamma$-ray one.

  2. 2.

Short-timescale correlation between $\gamma$-ray and optical light curves in
year one.

  3. 3.

Short-timescale correlation between $\gamma$-ray and mm-wave light curves in
year two.

  4. 4.

No measurable correlation between mm-wave and optical light curves.

These results have implications for the production mechanism of $\gamma$-rays
in blazars and the structure of these systems in general. Very broadly, the
correlated variability we observe between the $\gamma$-ray light curves and
those in the optical and mm-wave is more consistent with leptonic models of
$\gamma$-ray production than with hadronic models. While a quantitative
comparison of our findings with predictions of specific leptonic models are
beyond the scope of this paper, we note that simple scaling arguments predict
that, in the external inverse-Compton model, the fractional amplitude of a
$\gamma$-ray flare should scale linearly with the synchrotron flare amplitude.
On the other hand, in the synchrotron self-Compton model, the $\gamma$-ray
flare amplitude should be roughly the square of that seen in synchrotron. The
long-timescale flare in year one is of similar fractional amplitude in the mm-
wave and $\gamma$-ray data, lending some support to the external model. We
also note the longer lifetime of the mm-wave outburst is consistent with a
longer radiative lifetime of mm-wave electrons as discussed in, e.g., Potter
(2018).

Independent of the production of $\gamma$-rays, a puzzling feature of our data
is the complete lack of correlation between the optical and mm-wave data. For
year one, a possible explanation for this lack of correlation is that we are
seeing different regions of the jet in the two bands because the mm-wave
synchrotron radiation is optically thick. This would also be consistent with
the long-timescale correlation between the mm-wave and $\gamma$-ray light
curves and the short-timescale correlation of $\gamma$-ray and optical light
curves, because we would expect to see short-timescale variability only closer
to the central black hole. In this picture, the short-timescale mm-wave and
$\gamma$-ray correlation in year two is consistent with the mm-wave radiation
becoming optically thin. This motivates the comparison of mm-wave optical
thickness at different points in the light curve. To explore this, we extract
95 GHz SPTpol fluxes for PKS 2326-502 in a subset of observations, using
procedures identical those used to extract the 150 GHz flux (Section 2.1). We
measure the mm-wave spectral index which we define through:

$S_{\nu}\propto\nu^{\alpha};$ (6)

i.e., we estimate $\alpha$ as:

$\alpha=\frac{\ln(\frac{S_{150}}{S_{95}})}{\ln(\frac{\nu_{150}}{\nu_{95}})},$
(7)

where $S_{95}$ and $S_{150}$ are the 95 and 150 GHz fluxes, and $\nu_{95}$ and
$\nu_{150}$ are the effective band centers for a synchrotron source. We
estimate $\alpha$ for four 36-hour bundles each near the three prominent
features in the mm-wave light curve: the peak of the long-timescale flare in
year one, the short-timescale flare in year two, and the quiescent period in
year three.

We find values of the mm-wave spectral index of $\alpha=-0.24$ at the peak of
the year-one flare, $\alpha=-0.52$ in the year-two flare, and $\alpha=-0.95$
in the quiescent period. These values are consistent with the picture of the
mm-wave optical thickness decreasing after the year-one flare peak, thus
allowing us to see farther upstream in the mm-wave in year two than in year
one. What this scenario does not explain is why we do not see any correlation
between the optical and mm-wave radiation in year two. It is possible that the
optical synchrotron radiation tracing this activity is too faint, and that any
variation in the optical flux in year two is caused by an unassociated
process.

##  6 Conclusion

We have presented results from a pilot study using CMB data to monitor AGN, in
particular the blazar PKS 2326-502. We have correlated the mm-wave light curve
from SPTpol with $\gamma$-ray data from Fermi-LAT and optical data from
SMARTS. We measured long- and short-timescale correlation between the mm-wave
and $\gamma$-ray light curves, and short-timescale correlation between the
optical and $\gamma$-ray light curves, but we found no measurable correlation
between the mm-wave and optical light curves. These results are broadly
consistent with leptonic models of $\gamma$-ray production in blazars, but
they imply that the production of synchrotron emission is more complex than a
single source at all wavelengths.

While this study only used data from a single object, we have mm-wave data
from many more AGN in the SPTpol survey that we will use in future
investigations of multi-wavelength correlation. We will further expand this
monitoring program using the yet more sensitive data from the current camera
on the SPT, SPT-3G (Sobrin et al., 2022). Future experiments such as Simons
Observatory (Simons Observatory Collaboration et al., 2019) and CMB-S4 (CMB-S4
Collaboration et al., 2019) will cover up to 70% of the sky at nearly daily
cadence with similar or even higher sensitivity. These large-footprint, high-
cadence CMB surveys will be particularly well-suited for correlation with
optical monitoring from VRO-LSST (Ivezić et al., 2019). CMB experiments are
poised to become an integral part of the AGN monitoring landscape.

##  7 Acknowledgements

The South Pole Telescope program is supported by the National Science
Foundation (NSF) through awards OPP-1852617 and OPP-2147371. Partial support
is also provided by the Kavli Institute of Cosmological Physics at the
University of Chicago. John Hood acknowledges support from the NSF through
award OPP-2219065.

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# Movable Antenna-Empowered AirComp

Zhenqiao Cheng, Nanxi Li, Jianchi Zhu, Xiaoming She, Chongjun Ouyang, and Peng
Chen Z. Cheng, N. Li, J. Zhu, X. She, and P. Chen are with the 6G Research
Centre, China Telecom Beijing Research Institute, Beijing, 102209, China
(e-mail: {chengzq, linanxi, zhujc, shexm, chenpeng11}@chinatelecom.cn).C.
Ouyang is with the School of Information and Communication Engineering,
Beijing University of Posts and Telecommunications, Beijing, 100876, China
(e-mail: DragonAim@bupt.edu.cn).

###### Abstract

A novel over-the-air computation (AirComp) framework, empowered by the
incorporation of movable antennas (MAs), is proposed to significantly enhance
computation accuracy. Within this framework, the joint optimization of
transmit power control, antenna positioning, and receive combining is
investigated. An efficient method is proposed to tackle the problem of
computation mean-squared error (MSE) minimization, capitalizing on the
approach of alternating optimization. Numerical results are provided to
substantiate the superior MSE performance of the proposed framework, which
establish its clear advantage over benchmark systems employing conventional
fixed-position antennas (FPAs).

###### Index Terms:

Antenna position, mean-squared error (MSE), movable antenna (MA), over-the-air
computation (AirComp).

##  I Introduction

In traditional multiple-antenna systems, antennas remain stationary, which
imposes limitations on their capacity to fully exploit the spatial variations
within a specified transmit/receive region. This limitation becomes
particularly pronounced when dealing with a limited number of antennas [1]. In
response to this constraint, the concept of movable antennas (MAs) has been
introduced [1]. MAs are designed to overcome the constraints associated with
conventional fixed-position antennas (FPAs) [1, 2, 3, 4, 5]. They achieve this
by connecting to radio frequency (RF) chains via flexible cables and
incorporating real-time position adjustment mechanisms [1, 2, 3, 4, 5]. This
newfound flexibility empowers MAs to dynamically adapt their positions,
thereby reshaping the wireless channel to achieve superior wireless
transmission capabilities [1, 2, 3, 4, 5].

Existing research efforts have predominantly focused on exploring the
advantages of employing MAs to enhance communication performance [1, 2, 3, 4].
It is pertinent to acknowledge that the additional spatial degrees of freedom
afforded by antenna position optimization can yield benefits not only in terms
of throughput within communication networks but also in improving data
aggregation accuracy within computation networks. Motivated by this broader
perspective, our paper delves into the advantages derived from the integration
of MAs within over-the-air computation (AirComp) systems. In AirComp systems,
each sensor node (SN) is responsible for collecting a single sample of a
parameter of interest, such as temperature or humidity. These samples are then
transmitted over the wireless channel to a central data fusion center (FC) for
subsequent computation [6].

In our investigation, we seek to comprehensively understand how the
utilization of MAs can enhance the efficiency and accuracy of these
computation processes within AirComp systems. The main contributions of this
paper are summarized as follows: i) We propose an MA-empowered uplink AirComp
framework that leverages MAs to optimize antenna positions, leading to notable
improvements in computation mean-squared error (MSE). ii) We propose an
efficient alternating optimization algorithm to tackle the joint design of
antenna positioning, receive combining, and power control within the proposed
framework. iii) Numerical results reveal that the antenna placement gain
offered by MAs significantly improves the MSE performance than conventional
FPAs.

Figure 1: The MA-empowered AirComp system

##  II System Model

We consider uplink transmission in an MA-empowered AirComp setting where $K$
SNs send their collected data simultaneously to an FC for aggregation, as
depicted in Figure 1. The FC has $N$ receive MAs and each SN
$k\in{\mathcal{K}}\triangleq\\{1,\ldots,K\\}$ has a single transmit FPA. The
MAs are connected to RF chains via flexible cables, and thus their positions
can be adjusted in real time. The positions of the $n$th MA is denoted by
Cartesian coordinates
${\mathbf{r}}_{n}=[x_{n},y_{n}]^{\mathsf{T}}\in{\mathcal{C}}$ for
$n\in{\mathcal{N}}\triangleq\\{1,\ldots,N\\}$, where $\mathcal{C}$ denotes the
given two-dimensional receive region within which the MAs can move freely.
Without loss of generality, we set $\mathcal{C}$ as square regions with size
$A\times A$.

We assume quasi-static block-fading channels, and focus on one particular
fading block with the multi-path channel components at any location in
$\mathcal{C}$ given as fixed. The SN $k$-to-FC channel vector is denotes as
${\mathbf{h}}_{k}\in{\mathbbmss{C}}^{N\times 1}$. We consider the field-
response based channel model given by
${\mathbf{h}}_{k}=[h_{k}({\mathbf{r}}_{1})\ldots
h_{k}({\mathbf{r}}_{N})]^{\mathsf{T}}$, where [2, 3, 4, 5]

$\displaystyle
h_{k}({\mathbf{r}})\triangleq\sum\nolimits_{\ell=1}^{L_{k}}\sqrt{\mu_{k}}\sigma_{k,\ell}{\rm{e}}^{-{\rm{j}}\frac{2\pi}{\lambda}{\mathbf{r}}^{\mathsf{T}}{\bm{\rho}}_{k,\ell}},~{}\forall
k,$ (1)

and where
${\bm{\rho}}_{\ell,k}=[\sin{\theta_{\ell,k}}\cos{\phi_{\ell,k}},\cos{\theta_{\ell,k}}]^{\mathsf{T}}$,
$L_{k}$ is the number of channel paths, $\theta_{\ell,k}$ and $\phi_{\ell,k}$
are the elevation and azimuth angles of the $\ell$th path, respectively,
$\mu_{k}$ is the path loss, $\sigma_{k,\ell}$ denotes the small-scale fading,
and $\lambda$ is the wavelength. To characterize the performance limit, the FC
is assumed to have perfect channel state information [2, 3, 4].

At a specific time slot, we denote $x_{k}\in{\mathbbmss{C}}$ as the parameter
collected by SN $k$, which satisfies ${\mathbbmss{E}}\\{x_{k}\\}=0$,
${\mathbbmss{E}}\\{\lvert x_{k}\rvert^{2}\\}=1$, and
${\mathbbmss{E}}\\{x_{k}x_{k^{\prime}}\\}=0$, $\forall k\neq k^{\prime}$. Each
SN $k$ linearly scales its parameter by a power control factor, $w_{k}$, and
sends $w_{k}x_{k}$ to the FC simultaneously [6]. The FC exploits a linear
detector
${\mathbf{u}}=[u_{1},\ldots,u_{N}]^{\mathsf{T}}\in{\mathbbmss{C}}^{N\times 1}$
to combine the received signal and obtain the computing output of sum of the
gleaned parameters $x=\sum_{k=1}^{K}x_{k}$, which is expressed as follows: [6]

$\displaystyle\hat{x}={\mathbf{u}}^{\mathsf{H}}\Big{(}\sum\nolimits_{k=1}^{K}{\mathbf{h}}_{k}w_{k}x_{k}+{\mathbf{n}}\Big{)},$
(2)

where ${\mathbf{n}}\sim{\mathcal{CN}}({\mathbf{0}},\sigma^{2}{\mathbf{I}})$ is
the thermal noise at the FC with $\sigma^{2}$ being the noise power. The
computation distortion is measured by the estimation MSE of $x$, and is given
as [6]

$\displaystyle{\mathsf{MSE}}={\mathbbmss{E}}\\{\lvert{x}-\hat{x}\rvert^{2}\\}=\sum\nolimits_{k=1}^{K}\lvert{\mathbf{u}}^{\mathsf{H}}{\mathbf{h}}_{k}w_{k}-1\rvert^{2}+\sigma^{2}\lVert{\mathbf{u}}\rVert^{2}.$

Note that different from the FPA-based AirComp, the MSE for the MA-based
AirComp depends on the MA positions.

##  III Solution for MSE Minimization

We aim to minimize the MSE by optimizing the MA positions. The optimization
problem is formulated as

$\displaystyle\min\nolimits_{{\mathbf{r}}_{1},\ldots,{\mathbf{r}}_{N},{{w}}_{1},\ldots,w_{K},{\mathbf{u}}}~{}{\mathsf{MSE}}$
(3a) $\displaystyle~{}{\rm{s.t.}}~{}\lvert w_{k}\rvert^{2}\leq p_{k},\forall
k\in{\mathcal{K}},$ (3b)
$\displaystyle~{}\quad~{}~{}{\mathbf{r}}_{n}\in{\mathcal{C}},\forall
n\in{\mathcal{N}},\lVert{\mathbf{r}}_{n}-{\mathbf{r}}_{n^{\prime}}\rVert\leq
D,n\neq n^{\prime},$ (3c)

where $p_{k}$ denotes the power budget of SN $k$, $D$ is the minimum distance
required between each pair of antennas to avoid the coupling effect [2].
Problem (3) is challenging to solve due to the non-convex constraints and
tightly coupled variables. To this end, we partition the variable set into
$K+N+1$ distinct blocks
$\\{\mathbf{u}\\}\cup\\{w_{k}\\}_{k=1}^{K}\cup\\{\mathbf{r}_{n}\\}_{n=1}^{N}$.
Then, $K+N+1$ subproblems are solved in the sequel, which respectively
optimize $\mathbf{u}$, $w_{k}$, and $\mathbf{r}_{n}$, with all the other
variables being fixed. The developed algorithm can obtain a locally optimal
solution for (3) by iteratively solving the above subproblems in an
alternating manner.

Given $\\{w_{k}\\}_{k=1}^{K}\cup\\{\mathbf{r}_{n}\\}_{n=1}^{N}$, the optimal
solution of $\mathbf{u}$ is obtained by solving
$\frac{\partial{\mathsf{MSE}}}{\partial{\mathbf{u}}}={\mathbf{0}}$ as

$\displaystyle{\mathbf{u}}^{\star}=\Big{(}\sum\nolimits_{k=1}^{K}\lvert
w_{k}\rvert^{2}{\mathbf{h}}_{k}{\mathbf{h}}_{k}^{\mathsf{H}}+\sigma^{2}\mathbf{I}\Big{)}^{-1}\Big{(}\sum\nolimits_{k=1}^{K}w_{k}\mathbf{h}_{k}\Big{)}.$
(4)

Then, given $\\{\mathbf{u}\\}\cup\\{w_{k^{\prime}}\\}_{k^{\prime}\neq
k}\cup\\{\mathbf{r}_{n}\\}_{n=1}^{N}$, the marginal problem for $w_{k}$ is
expressed as ${\mathbf{w}}_{k}^{\star}=\operatorname*{argmin}\nolimits_{\lvert
w_{k}\rvert^{2}\leq
p_{k}}\lvert{\mathbf{u}}^{\mathsf{H}}{\mathbf{h}}_{k}w_{k}-1\rvert^{2}$. This
is a standard convex quadratic optimization problem whose solution is given by
$w_{k}^{\star}=\frac{{\mathbf{h}}_{k}^{\mathsf{H}}{\mathbf{u}}}{\lambda+\lvert{\mathbf{u}}^{\mathsf{H}}{\mathbf{h}}_{k}\rvert^{2}}$.
If $\lvert{\mathbf{u}}^{\mathsf{H}}{\mathbf{h}}_{k}\rvert^{2}\geq p_{k}^{-1}$,
we have $\lambda=0$; otherwise, we have
$\lambda=\frac{\lvert{\mathbf{u}}^{\mathsf{H}}{\mathbf{h}}_{k}\rvert}{\sqrt{p_{k}}}-\lvert{\mathbf{u}}^{\mathsf{H}}{\mathbf{h}}_{k}\rvert^{2}$.
The marginal problem for ${\mathbf{r}}_{n}$ is expressed as follows:

$\displaystyle\max_{{\mathbf{r}}_{n}\in{\mathcal{S}}_{n}}f_{n}\\!\triangleq\\!\sum\nolimits_{k=1}^{K}(2\Re\\{h_{k}^{*}({\mathbf{r}}_{n})c_{k,n}\\}\\!-\\!d_{k,n}\lvert
h_{k}({\mathbf{r}}_{n})\rvert^{2}),$ (5)

where $c_{k,n}=u_{n}w_{k}^{*}-\lvert w_{k}\rvert^{2}\sum_{n^{\prime}\neq
n}u_{n}u_{n^{\prime}}^{*}h_{k}({\mathbf{r}}_{n^{\prime}})$, $d_{k,n}=\lvert
w_{k}u_{n}\rvert^{2}$, and
${\mathcal{S}}_{n}\triangleq\\{{\mathbf{x}}|{\mathbf{x}}\in{\mathcal{C}},\lVert{\mathbf{x}}-{\mathbf{r}}_{n^{\prime}}\rVert\leq
D,\forall n\neq n^{\prime}\\}$. Due to the intractability of $f_{n}$,
stationary points of subproblem (5) can be found capitalizing on the gradient
decent method, whose details can found in [4, Section III-A3]. Furthermore,
following the steps outlined in [4], it is readily shown that our proposed
algorithm involves low computational complexity, which is of a polynomial
order.

##  IV Numerical Results

In this section, numerical results are provided to validate the performance of
the proposed MA-empowered AirComp system. Unless otherwise specified, we set
$K=4$, $p_{k}=p$, $\mu_{k}=-100$ dB, $\sigma_{k}^{2}=-100$ dBm, $L_{k}=4$,
$\sigma_{k,\ell}\sim{\mathcal{CN}}(0,L_{k}^{-1})$ ($\forall k,\ell$), and
$D=\frac{\lambda}{2}$. The elevation and azimuth angles are randomly set
within $[0,\pi]$. We compare the performance of our proposed algorithm with
the FPA-based benchmark scheme, where the FC is equipped with an FPA-based
uniform linear array with $N$ antennas spaced by $\frac{\lambda}{2}$.

Figure 2: MSE versus $p$.

Figure 3: MSE versus $A/\lambda$.

In Figure 3, we illustrate the MSE of the proposed and benchmark schemes
versus the transmit power $p$. It is observed that with the same power, our
proposed algorithm can achieve a lower MSE as compared to the schemes with
FPAs. Particularly, when achieving the same MSE, the MA-based scheme is
superior to its FPA-based counterpart by a power margin of approximately 3 dB.
The MSE performance gain mainly arises from the optimization of MA positions.

Figure 3 plots the MSE versus the normalized region size $A/\lambda$. It is
observed that the proposed schemes with MAs outperform FPA systems in terms of
computation distortion, and the performance gain increases with the region
size. It is also observed that the MA-based scheme exhibits convergence when
the value of $A/\lambda$ exceeds 4. This observation suggests that optimal MSE
performance for MA-empowered AirComp systems can be attained within a finite
receive region.

##  V Conclusion

In this paper, we proposed a novel MA-empowered AirComp system to reduce
computation distortion. We provided an efficient MSE minimization solution in
an MA-based uplink channel by optimizing the receive antenna positions.
Simulation results validated the superiority of the proposed MA-based
architecture over existing FPA-based ones.

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