# Polarized signatures of adiabatically expanding hotspots in Sgr A*’s
accretion flow

Joseph M. Michail,1,2 Farhad Yusef-Zadeh1,2, Mark Wardle,3 and Devaky
Kunneriath4
1Department of Physics and Astronomy, Northwestern University, 2145 Sheridan
Rd, Evanston, IL 60208, USA
2Center for Interdisciplinary Exploration and Research in Astrophysics
(CIERA), Northwestern University, 1800 Sherman Ave, Evanston, IL 60201, USA
3Research Centre for Astronomy, Astrophysics and Astrophotonics and School of
Mathematics and Physical Sciences, Macquarie University,
Sydney, NSW 2109, Australia
4National Radio Astronomy Observatory, 520 Edgemont Road, Charlottesville, VA
22903, USA<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We report 235 GHz linear and circular polarization (LP and CP) detections of
Sgr A* at levels of $\sim 10\%$ and $\sim-1\%$, respectively, using ALMA. We
describe the first full-Stokes modeling of an observed submillimeter flare
with an adiabatically-expanding synchrotron hotspot using a polarized
radiative transfer prescription. Augmented with a simple full-Stokes model for
the quiescent emission, we jointly characterize properties of both the
quiescent and variable components by simultaneously fitting all four Stokes
parameter light curves. The hotspot has magnetic field strength $71$ G, radius
$0.75$ Schwarzschild radii, and expands at speed $0.013$c assuming magnetic
equipartition. The magnetic field’s position angle projected in the plane-of-
sky is $\approx 55^{\circ}$ East of North, which previous analyses reveal as
the accretion flow’s angular momentum axis and further supports Sgr A* hosting
a magnetically-arrested disk. The magnetic field is oriented approximately
perpendicular to the line of sight, which suggests repolarization as the cause
of the high circular-to-linear polarization ratio observed at radio
frequencies. We additionally recover several properties of the quiescent
emission, consistent with previous analyses of the accretion flow, such as a
rotation measure $\approx-4.22\times 10^{5}$ rad m-2. Our findings provide
critical constraints for interpreting and mitigating the polarized variable
emission in future Event Horizon Telescope images of Sgr A*.

###### keywords:

Galaxy: centre – techniques: photometric – techniques: polarimetric – stars:
individual: Sgr A* – techniques: interferometric

††pubyear: 2023††pagerange: Polarized signatures of adiabatically expanding
hotspots in Sgr A*’s accretion flow–A

##  1 Introduction

Sagittarius A* (Sgr A*) is the $(4.152\pm 0.014)\times
10^{6}~{}\rm{M}_{\odot}$ supermassive black hole located at the Galactic
Center at a distance of $8178$ pc from the Earth (Gravity Collaboration et
al., 2019). Sgr A* is well-known to vary across the electromagnetic spectrum
(e.g., Yusef-Zadeh et al., 2011; Neilsen et al., 2013; Subroweit et al., 2017;
Do et al., 2019; Witzel et al., 2021; Wielgus et al., 2022b). In the radio and
submillimeter (submm), the emission is dominated by two components, both of
which originate from the accretion flow: (quasi-)quiescent and variable
radiation. The accretion flow is turbulent, which causes variations in its
flux on timescales $\gtrsim$ minute. The accretion flow produces an overall
net level of quiescent emission on top of low- and high-level amplitude
changes owing to its variable nature. The large-amplitude variations are known
as “flares” and dominate the variable emission. Previous work on Sgr A*’s
variability in the radio/submm regimes has focused on total intensity
observations. However, these earlier results largely ignore the polarization
of Sgr A*, which helps uncover the accretion flow’s magnetic properties.

Bower et al. (1999a, c, 2001) first searched for linear polarization (LP)
between 5-112 GHz and found Sgr A* to be linearly unpolarized at these
frequencies. However, circular polarization (CP) was detected at 5 and 8 GHz
(Bower et al., 1999b; Sault & Macquart, 1999). Further observations extended
polarimetric wavelength coverage from 1.5 GHz to 230 GHz (Bower et al., 2002;
Tsuboi et al., 2003; Muñoz et al., 2012). Aitken et al. (2000) first hinted at
intrinsic LP from Sgr A* at 400 GHz; however, their measurements may have been
contaminated by the abundance of polarized dust immediately surrounding Sgr A*
in the circumnuclear disk (e.g., Hsieh et al., 2018).

The first interferometric observations detected LP at 230 and 340 GHz (Bower
et al., 2003; Marrone et al., 2006). Later observations broadened this range
from 86 to 700 GHz (Macquart et al., 2006; Liu et al., 2016a, b). Presently,
Sgr A* is known to be circularly polarized from 1.5-230 GHz and linearly
polarized between 86-700 GHz. Liu et al. (2016a, b) find Sgr A*’s LP percent
to increase from $\sim 0.5\%$ at 93 GHz to $\sim 8.5\%$ at 500 GHz, which may
decrease at higher frequencies. Muñoz et al. (2012) compiled CP measurements
of Sgr A* from 1.5 GHz to 345 GHz finding levels of $\sim-0.2\%$ to
$\sim-1\%$, respectively. While the absolute CP amplitude is known to vary
(see Muñoz et al., 2012), the sign is consistently negative in the radio and
submm in all currently published data, which they suggest is caused by a
highly coherent magnetic field configuration throughout the accretion flow.

In addition to studying Sgr A*’s long-term polarimetric trends (i.e., Bower et
al., 2002, 2005; Macquart et al., 2006; Muñoz et al., 2012; Bower et al.,
2018), the detection of hourly-timescale variation of LP by Marrone et al.
(2006) opened up an additional avenue by which to study the accretion flow via
the variable emission. Several models describing the total intensity flaring
emission have been proposed, such as jets/outflows (Falcke & Markoff, 2000;
Brinkerink et al., 2015) and adiabatically-expanding synchrotron hotspots
embedded in the accretion flow (van der Laan, 1966; Yusef-Zadeh et al., 2006).
Yusef-Zadeh et al. (2007) first modeled the full-Stokes light curves of these
hotspots using an analytic formalism for the transfer of polarized synchrotron
radiation through a homogeneous medium (Jones & O’Dell, 1977a). Supplementing
this simple picture with full-Stokes radiative transfer presents a new
opportunity to study the magnetic field configuration in a localized region of
the accretion flow. Previously, only the equipartition magnetic field strength
could be estimated from this model. However, the observed polarization light
curves are regulated by the orientation of the magnetic field relative to the
observer. The orientation is a crucial physical parameter that could not
previously be determined. Yusef-Zadeh et al. (2007) tested this full-
polarization hotspot model at 22 and 43 GHz; however, their analysis was
limited as the LP level was low ($\sim 0.2-0.8\%$), and the data were noisy.
Sgr A* is brighter and more linearly and circularly polarized at submm
frequencies, which decrease the overall uncertainty in the polarimetric
properties.

In this paper, we present the first full-Stokes modeling of Sgr A*’s submm
flaring emission using the adiabatically-expanding hotspot model. This paper
is organized as follows. In Section 2, we discuss the observations and
processing of the data and analyze possible systematic issues in the CP
products. In Section 3, we describe the models adopted for the quiescent and
variable components used to fit the full-Stokes light curves and present the
best-fit values. For the first time, we determine the orientation of the
hotspot’s magnetic field on the plane-of-sky and along the line-of-sight. We
find the projected magnetic field to be oriented along the Galactic Plane and
approximately perpendicular to the line-of-sight. This has interesting
implications for the accretion flow’s magnetic field configuration, which we
discuss in Section 4. Furthermore, in Section 4, we discuss the other results
in the context of previous analyses in the literature and consider some
limitations with our chosen data set. Finally, in Section 5, we present a
summary of our findings and discuss future work.

##  2 Data

###  2.1 Observations and Processing

The Atacama Large Millimeter/submillimeter Array (ALMA) observed Sgr A* on 16
July 2017 in band 6 ($\approx 230$ GHz) in full polarization (project ID
2016.A.00037.T). These data, part of a multi-wavelength campaign of Sgr A*
concurrent with the Chandra X-ray Observatory and the Spitzer Space Telescope,
were taken with the 12-meter array in the C40-5 configuration (the baselines
ranged from 17 to 1100 meters). For our analysis, we focus only on the submm
data.

The observation consists of two line and two continuum spectral windows. The
two continuum windows are centered on 233 and 235 GHz, each having a bandwidth
of 2 GHz with 64 31.25-MHz bandwidth channels. The first spectral line window
is centered on SiO (5-4) at $\approx$217 GHz with a 1.875 GHz bandwidth of
1920 0.976-MHz channels. The second spectral line window is centered on 13CO
(2-1) at 220.398 GHz with 1920 0.244-MHz channels for a total bandwidth of
$\approx 0.47$ GHz. In this analysis, we average over all of the channels per
spectral window to obtain four frequency-averaged continuum windows.

Only one of five execution blocks was observed owing to technical issues which
occurred during the observation. We used the ALMA pipeline (version
2020.1.0.40) with CASA 6.1.1.15 (McMullin et al., 2007) to generate the
calibrated data. The following calibrators are used to generate the
calibration tables: J1733-1304 (flux), J1517-2422 (bandpass), J1549+0237
(instrumental polarization), and J1744-3116 (phase). The QA2 team designated
these data “semi-pass” since the parallactic coverage ($\approx 46^{\circ}$)
was lower than recommended to determine the instrumental polarization terms
($60^{\circ}$). Despite this, we were able to calibrate the instrumental
polarization. We imaged and phase self-calibrated the data starting with a
solution interval of $30$ seconds and stopping at a single integration time
($6.05$ seconds). After phase self-calibration, we flagged any obvious
misbehaving baselines or antennas.

Figure 1: A sample of example images of Sgr A* on 16 July 2017 at 00:28:07 UTC
using a 30-second binning time for each Stokes parameter at 233.5 GHz. The
full image is $\approx 51\arcsec$ per side. We flag pixels below a normalized
primary beam limit of 20%, resulting in an image that is approximately
$37\arcsec$ per side. The panels use the same gray scale to show the noise
level. The inset is a $2\aas@@fstack{\prime\prime}5\times
2\aas@@fstack{\prime\prime}5$ subregion centered on Sgr A*.

We developed a CASA script to autonomously determine the full polarization
light curves for a point source located at the phase center using TCLEAN and
IMFIT. Briefly, the code bins every scan on a single source to a user-defined
value for imaging. TCLEAN images the visibilities in all four Stokes
parameters for each time bin. IMFIT is used to fit a point source + zero-level
offset at the phase center, where Sgr A* is located, in each image and
polarization to determine the point source flux density and (statistical)
error. We construct the point source light curves using the fitted IMFIT
parameters and export them to a text file for further analysis, where we
calculate the polarization product light curves (see Appendix A for our chosen
conventions). Since Sgr A* is surrounded by diffuse emission which is not
fully resolved out in the observed configuration, this method yields
contamination-free light curves without restricting the projected baseline
length, which would lower the overall sensitivity.

We use 30-second binning in our analysis. Each image is $1024\times 1024$
($\approx 51\arcsec\times 51\arcsec$) and uses a cell size of
$0\aas@@fstack{\prime\prime}05$. We apply the standard 20% primary beam cut to
remove imaging artifacts toward the edge of the image, resulting in a final
image size of $\approx 37\arcsec\times 37\arcsec$. We do not primary-beam
correct the image since Sgr A* is at the phase center. We restrict the maximum
number of iterations to $1000$ to properly clean any extended emission while
not cleaning noise artifacts. We show a sample image of Sgr A* during a
30-second binning time in all four Stokes parameters in Figure 1. In Figure 2,
we show the final Stokes I, Q, U, and V, LP percent ($p_{l}$), CP percent
($p_{c}$), and LP angle ($\chi$) light curves used in our analysis. Overall,
we find Sgr A* to be linearly and circularly polarized at levels of $\sim
10\%$ and $\sim-1\%$, respectively. The definitions of these parameters and
their uncertainties are detailed in Appendix A. We discuss the absence of CP
products for J1744-3116 in Section 2.2.

###  2.2 Verifying Sgr A*’s Circular Polarization Detection

There has been great care taken in previous polarimetric analyses to rule out
calibration-error-based CP detections. Goddi et al. (2021) present a detailed
description of the issue (see their Appendix G). In short, the polarization
calibrator is assumed to have Stokes $V=0$, which can induce a false (and
time-dependent) CP onto the target sources. To check for systematics, we focus
on the CP characteristics of J1517-2422 and J1744-3116 following the
prescription given in Muñoz et al. (2012). J1517-2422 has a similar
declination to Sgr A*
($17^{\rm{h}}45^{\rm{m}}40.04^{\rm{s}},-29^{\circ}00\arcmin 28.17\arcsec$),
while J1744-3116 has a comparable right ascension. To check for intrinsic CP
for the calibrators, we image each spectral window in Stokes I and V during
the entire observing window using the same non-interactive process in Section
2.1. We obtain a higher sensitivity image to detect CP by imaging the entire
observation. We fit a point source to the phase center using IMFIT and report
the integrated flux density and statistical error. These results are shown in
Table 1. While IMFIT returns converged flux densities and errors for
J1744-3116 in Stokes V, the images do not show circularly polarized emission
at or near the phase center. To quantify the $3\sigma$ upper limit on CP, we
calculate $3\times$ the Stokes V root-mean-square (RMS) provided by IMSTAT.

For J1517-2422, we detect a statistically significant $p_{c}\approx-0.1\%$.
Since this source is bright ($>3$ Jy), residual or uncalibrated instrumental
polarization terms in V could lead to spurious CP measurements. Despite the
unfortunately sparse coverage of this source, we compare our results to those
in the literature. Bower et al. (2018) report $p_{c}\approx 0.1\%$ for this
source in August 2016 at $\approx 240$ GHz, having the same magnitude (but
opposite sign) as our result. Following Goddi et al. (2021), we use the
AMAPOLA111http://www.alma.cl/~skameno/AMAPOLA/ project, which tracks the flux
density and polarization properties of several ALMA calibrators for more
nearby observations to July 2017. At 233 GHz, the CP of J1517-2422 ranged
between roughly $-0.4\%$ and $0.3\%$ during January–April 2017 and between
$-1.0\%$ to $-0.4\%$ between October–December 2017. Given that our $-0.1\%$
detection is well within the historical average and that Sgr A* is at least
$10\times$ more circularly polarized than either J1517-2422 or J1744-3116, we
robustly detect intrinsic CP from Sgr A*.

$\nu$ | Stokes I | Stokes V | $p_{c}$
---|---|---|---
$\rm{[GHz]}$ | [mJy] | [mJy] | [%]
J1517-2422
$217.1$ | $3099\pm 0.54$ | $-3.33\pm 0.17$ | $-0.11\pm 0.005$
$220.0$ | $3088\pm 0.59$ | $-2.37\pm 0.17$ | $-0.08\pm 0.006$
$233.5$ | $3059\pm 0.61$ | $-2.75\pm 0.17$ | $-0.09\pm 0.006$
$235.0$ | $3049\pm 0.56$ | $-2.16\pm 0.14$ | $-0.07\pm 0.005$
J1744-3116
|  |  |  $3\sigma$ Upper Limits
$217.1$ | $217.4\pm 0.17$ | $<\left|0.27\right|$ | $<|0.12|$
$220.0$ | $214.3\pm 0.29$ | $<|0.51|$ | $<|0.24|$
$233.5$ | $210.2\pm 0.15$ | $<|0.27|$ | $<|0.13|$
$235.0$ | $207.8\pm 0.17$ | $<|0.27|$ | $<|0.13|$
Table 1: The measured Stokes I and V properties for the two non-instrumental
polarization calibrators on 16 July 2017. The errors quoted are only
statistical. We do not detect CP in J1744-3116 but do detect it in J1517-2422
at a level $\approx-0.1\%$.

The final aspect to consider is a time-dependent Stokes V leakage. We cannot
directly check for this as J1744-3116 is not circularly polarized, and no
other gain calibrators were observed. Goddi et al. (2021, Appendix G) study
the measured Stokes V as a function of feed angle (parallactic angle +
receiver rotation relative to the antenna mount) in search for uncalibrated
Stokes V terms. In their April 2017 data (near 230 GHz), they found a
modulating $\approx 0.1\%$ leakage in Stokes V for the nearest calibrators to
Sgr A* (J1733-1304 and J1924-2914). This modulation occurs over a range of
$\gtrsim 100^{\circ}$ in the feed angle. In this observation, the feed angle
changes by only $\approx 8^{\circ}$. By our estimates, this induces a maximum
of $\approx 2$ mJy (absolute) variation in Stokes V. As Sgr A*’s Stokes V
light curves vary by $\approx 15$ mJy, these time-dependent variations are
intrinsic to Sgr A* and are not caused by uncalibrated polarization terms.

##  3 Modeling the Light Curves

Figure 2: Full Stokes and polarization light curves of Sgr A* (red, left) and
the phase calibrator J1744-3116 (blue, right) on 16 July 2017. As J1744-3116
is not circularly polarized (see Section 2.2), not shown are Stokes V and
$p_{c}$ for this source. Error bars for both sources are shown and are often
smaller than the marker size.

We adopt a two-component model consistent with previous work to account for
the variable and (quasi-)quiescent components of Sgr A*’s light curves. In
contrast to previous work, however, we incorporate a full-Stokes picture. The
flaring component is modeled as a homogeneous, spherical synchrotron hotspot
adiabatically expanding at a constant speed on a roughly one-hour timescale.
This model is characterized by several physical parameters, such as the
initial radius, expansion speed, magnetic field strength and orientation, and
power-law population of relativistic electrons. Our model does not
intrinsically include orbital motion (i.e., a varying magnetic field
orientation), gravitational effects (i.e., lensing), non-symmetric geometric
evolution (i.e., shearing), nor a sense of the hotspot’s location in the
accretion flow. We account for secular variations in the accretion flow by
modeling the slowly-varying frequency-dependent quiescent component. At each
frequency, the four Stokes parameters are assumed to rise or fall linearly
during the observation and are characterized by phenomenological parameters,
such as gradients with respect to time and reference flux densities.
Additionally, the Stokes parameters are frequency-dependent, accounting for
physical properties of the accretion flow (such as rotation measure, RM),
which we model with spectral indices and gradients with respect to frequency.
The two components are described in detail below.

###  3.1 Polarization Model for Flaring Emission

The Stokes I temporal- and frequency-dependent flaring emission are well-
modeled by an adiabatically-expanding synchrotron plasma (thenceforth referred
to as a “hotspot;” van der Laan, 1966; Yusef-Zadeh et al., 2006). The hotspot
is homogeneous and characterized by five parameters: $I_{p}$, $p$,
$v_{\rm{exp}}$, $R_{0}$, and $t_{0}$. $I_{p}$ is the peak flare flux density
at frequency $\nu_{0}$ at time $t_{0}$ having radius $R_{0}$, $p$ is the
electron energy power-law index $\left(N(E)\propto E^{-p}\right)$ valid
between energies $E_{\rm{min}}$ and $E_{\rm{max}}$, and $v_{\rm{exp}}$ is the
(normalized) radial expansion velocity. The flux density at any frequency and
size is calculated via

$\displaystyle
I_{f}\left(R\right)=I_{p}\left(\dfrac{\nu}{\nu_{0}}\right)^{5/2}\left(\dfrac{R}{R_{0}}\right)^{3}\dfrac{f(\tau_{\nu})}{f(\tau_{0})}.$
(1)

$f(...)$ is a non-trivial function encompassing the full-Stokes radiative
transfer equations (briefly described below). $\tau_{\nu}$, the frequency- and
size-dependent optical depth, is given by,

$\displaystyle\tau_{\nu}(R)=\tau_{0}\left(\dfrac{\nu}{\nu_{0}}\right)^{-(p+4)/2}\left(\dfrac{R}{R_{0}}\right)^{-(2p+3)}.$
(2)

$\tau_{0}$ is the critical optical depth where the hotspot becomes optically
thin and is determined by

$\displaystyle e^{\tau_{0}}-\left(\dfrac{2p}{3}+1\right)\tau_{0}-1=0.$ (3)

Finally, we assume a uniform expansion to relate the time and radius:

$\displaystyle R(t)=R_{0}\left(1+v_{\rm{exp}}\left(t-t_{0}\right)\right).$ (4)

Assuming magnetic equipartition, we can determine the hotspot’s physical
radius and expansion velocity, mass, magnetic field strength, and electron
number density.

Given the remarkable high-sensitivity light curves found in Section 2, we
model them using a full-Stokes adiabatically-expanding synchrotron hotspot. To
do so, we use the prescription given in Jones & O’Dell (1977a), which
describes the transfer of full-Stokes synchrotron radiation for a static
source through a homogeneous medium, i.e., where $v_{\rm{exp}}\ll c$. The
Stokes I model described above temporally evolves depending upon the size of
the emitting region. Since this temporal evolution is secular, we can model
the expanding source as a sequence of hotspots with varying parameters (such
as radius, magnetic field strength, and electron population) to account for
the time-dependent properties as it grows. Therefore, we convert a stationary
solution into a dynamic one to produce full-Stokes light curves of such a
source.

Supplementing this model with polarization adds only two parameters: $\theta$
and $\phi$, which are related to the orientation of the magnetic field. A
schematic of these angles is shown in Figure 3. $\phi$ is the intrinsic
electric vector position angle (EVPA; see Equation 16) of the hotspot,
measured East of North, projected in the plane-of-sky (POS). It is closely
related to the projected magnetic field orientation in the POS,
$\phi_{B}=\phi+\pi/2$. $\phi$ and $\phi_{B}$ are pseudovectors and obey the
$\pi$-ambiguity. $\theta$ is the projected magnetic field vector to the line
of sight (LOS). In this convention, $\theta=0^{\circ}$ and $\theta=90^{\circ}$
occur when the projected magnetic fields are along and perpendicular to the
LOS, respectively.

Under the assumption of homogeneity, the radiative transfer equations as given
by Jones & O’Dell (1977a) read

$\displaystyle\normalsize{\begin{bmatrix}\vspace{0.15cm}J_{\nu}\\\
\vspace{0.15cm}\epsilon_{Q}J_{\nu}\\\ \vspace{0.15cm}0\\\
\vspace{0.15cm}\epsilon_{V}J_{\nu}\end{bmatrix}=\begin{bmatrix}\left(\frac{d}{d\tau_{\nu}}+1\right)&\zeta_{Q}&0&\zeta_{V}\\\
\zeta_{Q}&\left(\frac{d}{d\tau_{\nu}}+1\right)&\zeta_{V}^{*}&0\\\
0&-\zeta_{V}^{*}&\left(\frac{d}{d\tau_{\nu}}+1\right)&\zeta_{Q}^{*}\\\
\zeta_{V}&0&-\zeta_{Q}^{*}&\left(\frac{d}{d\tau_{\nu}}+1\right)\\\
\end{bmatrix}\begin{bmatrix}\vspace{0.15cm}I_{f}/\Omega\\\
\vspace{0.15cm}Q_{f}/\Omega\\\ \vspace{0.15cm}U_{f}/\Omega\\\
\vspace{0.15cm}V_{f}/\Omega\end{bmatrix}}.$ (5)

Inside the dielectric tensor, $\tau_{\nu}$ is the optical depth,
$\zeta_{\\{Q,V\\}}$ are the Stokes Q and V absorption coefficients,
$\zeta^{*}_{V}$ is the rotativity coefficient (responsible for Faraday
rotation), and $\zeta^{*}_{Q}$ is the convertability (or repolarization;
Pacholczyk, 1973) coefficient between LP and CP. $\epsilon_{\\{Q,V\\}}$ are
the Stokes Q and V emissivity coefficients, and $J_{\nu}$ is a source
function. $\Omega$ is the solid angle subtended by the source given by
$\Omega\equiv\pi R^{2}/d^{2}$, where $R$ and $d$ are the radius of the hotspot
and its distance from the Earth, respectively. Jones & O’Dell (1977a) provide
analytic solutions for the emergent Stokes flux densities ($I_{f}$, $Q_{f}$,
$U_{f}$, and $V_{f}$) integrated over the homogeneous source (see their
equations C4-C17).

Figure 3: We show several perspectives of the various angles used in this
analysis. (a) The general schematic setup, where “N,” “E,” and “$\hat{k}$”
denote north, east (in equatorial coordinates), and the unit vector toward the
observer, respectively. The hotspot (gray sphere) possesses a three-
dimensional magnetic field vector ($\vec{B}$, orange). The pink dashed line
denotes the projected magnetic field orientation ($\phi_{B}$) in the North-
East plane perpendicular to the LOS, measured East of North. The dot-dashed
cyan arrow shows the angle between the projected magnetic field vector to the
LOS ($\theta$). (b) The schematic along the observer’s LOS. The dashed pink
line again shows the projected magnetic field orientation. In this analysis,
we focus on the electric vector position angle (EVPA, $\phi$) shown as a solid
green line. The EVPA is also measured East of North. $\phi_{B}$ and $\phi$ are
related by $\phi_{B}=\phi+\pi/2$ and is wrapped through the $\pi$-ambiguity.
For clarity, we do not show the dot-dashed cyan vector. (c) A “side” view
along the eastern direction to show the projected magnetic field vector along
the LOS. Again, for clarity, we do not show the dashed pink or solid green
lines.

We note the number of assumptions made in this picture, specifically that the
hotspot undergoes only secular evolution, which is a limitation in our
modeling and differs from other approaches (such as Tiede et al., 2020; Gelles
et al., 2021) that include the hotspot’s evolution as it orbits Sgr A*. Our
attempt to formally fit this data is to describe the general nature of the
expanding hotspot. However, if these secondary processes dominate adiabatic
expansion, we would not have expected this modeling to be successful. This is
due to the frequency- and polarization-dependent coupling of the polarized
radiative transfer equations (Equation 5 and see Section 4.4). These secondary
effects can be included and are planned for future work.

###  3.2 Quiescent Frequency and Temporal Variations

The quiescent component is known to have frequency- and time-dependent
baselines that must be accounted for while modeling the flaring emission. The
time dependence likely arises from continual, longer-term variability within
the accretion flow. For example, Dexter et al. (2014) find an $\sim 8$-hour
characteristic timescale in Sgr A*’s submm light curves. While their analysis
focuses only on Stokes I, we include time-dependent terms for the other three
Stokes parameters for consistency. If there is no time dependence in the
Stokes Q, U, and V light curves, we expect their time-dependent fitting
parameters to be consistent with $0$. Frequency-dependent variations emerge
from processes like optical depth (Stokes I and V) or RM (Stokes Q and U).

To account for the frequency- and time-dependent nature of the quiescent
component, we use the following model:

$\displaystyle I_{q}(\nu,t)$
$\displaystyle=\left(I_{0}+I_{1}\left(t-t_{0}\right)\right)\left(\dfrac{\nu}{\nu_{0}}\right)^{\alpha_{I}}\,,$
(6) $\displaystyle Q_{q}(\nu,t)$
$\displaystyle=Q_{0}+Q_{1}\left(t-t_{0}\right)+Q_{2}\left(\nu-\nu_{0}\right)\,,$
(7) $\displaystyle U_{q}(\nu,t)$
$\displaystyle=U_{0}+U_{1}\left(t-t_{0}\right)+U_{2}\left(\nu-\nu_{0}\right)\,,$
(8) $\displaystyle V_{q}(\nu,t)$
$\displaystyle=\left(V_{0}+V_{1}\left(t-t_{0}\right)\right)\left(\dfrac{\nu}{\nu_{0}}\right)^{\alpha_{V}}\,.$
(9)

We choose two different frequency dependencies based on the Stokes parameter.
The signs of Stokes I and V cannot or do not, respectively, change across the
18 GHz of bandwidth. (The sign of Stokes V does not change from $1.4-340$ GHz;
Muñoz et al., 2012). Therefore, we use the classic frequency power-law form.
Equations 6 and 9 follow the time- and frequency-dependent model in Michail et
al. (2021a). Due to Faraday rotation, the signs for Stokes Q and U change
owing to RM across the bandpass. Therefore, we model frequency-dependent
changes in Stokes Q and U using a linear form. Here, $I_{i},Q_{i},U_{i},$ and
$V_{i}$ are all constants; parameters with a “$0$” subscript reflect reference
flux densities for the four Stokes parameters at frequency $\nu_{0}$ at time
$t_{0}$. The time- and frequency-dependent slopes are denoted by parameters
with subscripts “$1$” and “$2$”, respectively. The spectral indices for Stokes
I and V are $\alpha_{I}$ and $\alpha_{V}$, respectively.

###  3.3 Results of Model Fitting

We use LMFIT (Newville et al., 2014) to simultaneously fit the 16 light curves
(4 spectral windows $\times$ 4 Stokes parameters) by minimizing the $\chi^{2}$
of the variable + quiescent models (i.e., $I_{\nu}=I_{f}+I_{q}$,
$Q_{\nu}=Q_{f}+Q_{q}$, $U_{\nu}=U_{f}+U_{q}$, $V_{\nu}=V_{f}+V_{q}$) discussed
above. Due to the lack of time coverage, we only fit the data through 00:36
hrs UTC. While there appears to be a second flare beginning near 00:45 hrs
UTC, the time coverage is insufficient to model it. Therefore, we do not
include those data in the fit. We discuss the implications of this limited
time coverage in Section 4.3. For this analysis, we set the reference
frequency to $\nu_{0}=235.1$ GHz. In Table 2, we present the fitted parameters
values and errors. In Figure 4 (left), we show the best-fit model superimposed
on Sgr A*’s light curves.

Parameter | Description | Value | Unit
---|---|---|---
Hotspot
$I_{p}$ | Peak Flare Flux Density at 235.1 GHz | $0.19\pm 0.01$ | Jy
$p$ | Electron Power-Law Index | $3.10\pm 0.05$ | –
$v_{\rm{exp}}$ | Relative Expansion Speed | $1.48\pm 0.05$ | hr-1
$\phi$ | Intrinsic EVPA projected in POS (East of North) | $144.8\pm 1.5$ | degrees
$\theta$ | Angle of projected magnetic field vector relative to LOS | $90.09\pm 0.01$ | degrees
$t_{0}$ | Time of Peak Flux at 235.1 GHz | $0.455\pm 0.002$ | hr UTC
Quiescent Component
$I_{0}$ | Stokes I flux density at $t=t_{0}$ at 235.1 GHz | $2.46\pm 0.01$ | Jy
$Q_{0}$ | Stokes Q flux density at $t=t_{0}$ at 235.1 GHz | $-82.6\pm 3.7$ | mJy
$U_{0}$ | Stokes U flux density at $t=t_{0}$ at 235.1 GHz | $-292.3\pm 4.7$ | mJy
$V_{0}$ | Stokes V flux density at $t=t_{0}$ at 235.1 GHz | $-16.5\pm 1.2$ | mJy
$I_{1}$ | Stokes I Time-Dependent Slope | $-238.2\pm 13.0$ | mJy hr-1
$Q_{1}$ | Stokes Q Time-Dependent Slope | $-118.3\pm 14.2$ | mJy hr-1
$U_{1}$ | Stokes U Time-Dependent Slope | $-146.6\pm 16.5$ | mJy hr-1
$V_{1}$ | Stokes V Time-Dependent Slope | $6.5\pm 5.2$ | mJy hr-1
$\alpha_{I}$ | Stokes I Spectral Index | $0.16\pm 0.01$ | –
$Q_{2}$ | Stokes Q Frequency-Dependent Slope | $2.82\pm 0.07$ | mJy GHz-1
$U_{2}$ | Stokes U Frequency-Dependent Slope | $-2.88\pm 0.09$ | mJy GHz-1
$\alpha_{V}$ | Stokes V Spectral Index | $-1.31\pm 0.45$ | –
Table 2: Fitted parameters (with errors) for the joint quiescent and variable
model discussed in Section 3.

Figure 4: Left: Sgr A*’s light curves are shown in red, and the best-fit model is superimposed in black. Due to the short time coverage, we only model the light curves before 00:36 hr UTC. Right: Light curves and linear polarimetric quantities for the quiescent component. The best-fit model is plotted in red, and the $1\sigma$ error range is shaded in gray. Unlike the left panel, these figures are plotted to 00:36 hr UTC. Parameter | Description | Value
---|---|---
Adopted Parameters
$E_{\rm{min}}$ | Electron Lower Energy Bound |  $1$ MeV
$E_{\rm{max}}$ | Electron Upper Energy Bound |  $500$ MeV
Derived Parameters
$n_{e}$ | Electron density |  $6.5\times 10^{7}$ cm-3
$R_{0}$ | Radius of flaring region at $t=t_{0}$ |  $9.2\times 10^{11}$ cm
$v_{\rm{exp}}\cdot R_{0}$ | Physical Expansion Speed |  $0.013$c
$B_{\rm{eq}}$ | Equipartition magnetic field strength |  $71$ G
$M$ | Mass of flaring region |  $3.62\times 10^{20}$ g
Table 3: Adopted and derived hotspot properties from the variable parameters
in Table 2. We do not account for non-relativistic electrons or protons while
estimating the equipartition magnetic field strength. Therefore, this is a
lower limit on the true value.

####  3.3.1 Variable Component

Modeling the light curves gives the six variable component parameters, which
characterize the hotspot and are listed in Table 2. To determine physical
parameters, we assume the hotspot is in magnetic equipartition with the
electrons responsible for the synchrotron emission between energies
$E_{\rm{min}}$ and $E_{\rm{max}}$. In Table 3, we present the physical
properties of the hotspot fixing $E_{\rm{min}}$ and $E_{\rm{max}}$ to $1$ and
$500$ MeV ($\gamma_{e}\sim 2-1000$), respectively. We disregard contributions
from protons and non-relativistic electrons in the magnetic field strength, so
this is a lower limit on the true value. Overall, we find a $235.1$ GHz peak
flare flux density of $0.19$ Jy produced by an electron energy spectrum
$N(E)\propto E^{-3.1}$. The hotspot expands at speed $\approx 0.013c$ with an
equipartition magnetic field strength $71$ G and radius $0.75~{}R_{\rm{S}}$
($1~{}R_{\rm{S}}=1.23\times 10^{12}$ cm for a $4.152\times 10^{6}~{}M_{\odot}$
Schwarzschild black hole). Our model robustly detects the two new parameters
in this full-polarization fit: $\theta$ and $\phi$. For the intrinsic EVPA of
the source, we find $\phi\approx 145^{\circ}$, corresponding to
$\phi_{B}=55^{\circ}$ East of North ($\phi+\pi/2$ wrapped through the
$\pi$-ambiguity). Additionally, we determine $\theta=90.09^{\circ}$, placing
the projected magnetic field orientation approximately perpendicular to the
LOS.

To compare the overall variability of the flaring component to the quiescent
emission, we calculate the hotspot’s mean LP and CP and their relative
fractional change
($\rm{RFC}\equiv\left(\rm{max}-\rm{min})/\rm{average}\right)$). During the
modeled range, we find the flare to have average LP and CP of $\approx 35\%$
and $\approx-4.2\%$, respectively, at $235.1$ GHz. The LP goes from a minimum
of $\approx 9.5\%$ to a maximum of $\approx 81\%$, giving an RFC $=~{}2.04$.
The CP ranges from $\approx-15\%$ to $\approx-0.1\%$ with an RFC $=~{}3.48$.

####  3.3.2 Quiescent Component

We find statistically-significant time dependencies in the quiescent
component’s Stokes I, Q, and U light curves. In Figure 4 (right), we present
the quiescent-only full-Stokes light curves during our modeled range. While
the Stokes I time-dependence has been observed previously (e.g., Michail et
al., 2021a), this is the first detection of the quiescent component’s Stokes Q
and U time-variability. We do not find any changes in the quiescent emission’s
Stokes V properties, as the time-dependent term is not significant. An
uncalibrated Stokes V polarization term (Section 2.2) would contribute to the
final fitted value, further proof that the variations in Stokes V are
intrinsic to Sgr A*’s flaring emission.

We find the quiescent emission has average LP $\approx 12\%$ and average CP of
$\approx-0.7\%$. The LP ranges between $\approx 9.9\%$ to $\approx 14\%$,
giving RFC $=~{}0.31$. Since we conclude above the Stokes V quiescent emission
is not time-dependent, we do not calculate its RFC.

Additionally, we identify strong frequency-dependent terms in Stokes I, Q, and
U, and a marginal dependence in Stokes V. We find the quiescent emission’s
Stokes I spectral index is $\alpha_{I}=0.15$. The Stokes V spectral index
value is much steeper at $\alpha_{V}=-1.31$. The detection of frequency-
dependent slopes in Stokes Q and U generates a non-zero RM in the quiescent
emission, which has been found in previous analyses (e.g., Marrone et al.,
2006; Bower et al., 2018).

##  4 Discussion

In the previous section, we presented the first-ever full-Stokes modeling of
Sgr A*’s total intensity and polarized light curves to simultaneously
characterize the quiescent and variable emission. Determining the fitted
parameters for our two-component model allows us to study and derive
additional physical properties of both components. We derived a few of these
properties for the variable emission above by assuming magnetic equipartition.
In this section, we compare our results for each component to previous
analyses and broadly find them consistent with those in the literature.
Finally, given our $\sim 40$ minute observation and the 18 free parameters in
this fit, we examine their implications on our results.

###  4.1 Variable Component

The electron power-law index responsible for the flaring emission is
consistent with multi-wavelength constrained values, which broadly range from
$p\approx 1-3$ (e.g., Yusef-Zadeh et al., 2006, 2008; Eckart et al., 2009;
Michail et al., 2021b; GRAVITY Collaboration et al., 2021; Witzel et al.,
2021). The calculated magnetic field strength is on the higher side of those
previously reported, which typically averages a few to tens of Gauss. However,
Yusef-Zadeh et al. (2008) and Eckart et al. (2009) report magnetic field
strengths $\approx 80$ G. These are somewhat stronger than the average field
strengths of stellar wind-fed simulations (e.g., Ressler et al., 2020). It
suggests the hotspot may have occurred near the inner accretion flow, where
field strengths are stronger and/or in a concentration of flux when the
acceleration of particles drives the flaring. The $\approx 7\times 10^{7}$
cm-3 electron density is consistent with the Witzel et al. (2021) joint
variability analysis of Sgr A* at submm, infrared, and X-ray frequencies.

Previous observations (i.e., Marrone et al., 2007; GRAVITY Collaboration et
al., 2018; Wielgus et al., 2022a) have detected variability in the LP angle
($\chi$) of Sgr A* caused by orbital motion. $\chi$ changes by $180^{\circ}$
over half of the orbital period of the hotspot (GRAVITY Collaboration et al.,
2018). We only observe $\Delta\chi\sim 15^{\circ}$ over $\sim 40$ minutes, the
latter of which roughly corresponds to the period at the innermost stable
circular orbit (ISCO) for a non-spinning black hole with the mass of Sgr A*
($P=31.5$ minutes). If the hotspot was near the ISCO, we expect
$\Delta\chi\sim 180^{\circ}$ from orbital motion during our modeling range,
which would dominate over changes caused by adiabatic expansion. However, the
observed change of $\sim 15^{\circ}$ implies the hotspot is far outside the
ISCO, and orbital motion-induced variations in $\chi$ are subdominant to
changes from the adiabatic expansion. Therefore, the $71$ G field we derive
above, which places us towards the inner accretion flow when compared to
Ressler et al. (2020), is likely overestimated. We discuss the possible cause
in Section 4.3.

We find the magnetic field angle projected on the POS is $\phi_{B}=55^{\circ}$
East of North. As a point probe of the conditions within the accretion flow,
these results present the first direct detection of the accretion flow’s
projected magnetic field orientation in the POS. Several analyses suggest this
position angle is a favored orientation for the Sgr A* system. Near-infrared
polarimetric observations of Sgr A*’s flaring emission over several years find
a mean EVPA of approximately $60^{\circ}$ East of North with a range of about
$45^{\circ}$ (Eckart et al., 2006; Meyer et al., 2007). Eckart et al. (2006)
speculate this indicates the projected spin axis of a disk around Sgr A*,
while Meyer et al. (2007) merely propose this as a preferred orientation for
the Sgr A* system. Continuum and spectral observations near 1.5 GHz by Yusef-
Zadeh et al. (2020) find a symmetric jet-like structure oriented along the
Galactic plane at a position angle $\sim 60^{\circ}$, which they attribute to
evidence of a jet/outflow from Sgr A*. A more recent analysis by Wielgus et
al. (2022a) uses ALMA linear polarimetry at similar frequencies ($\sim 220$
GHz) in the context of an orbiting, non-expanding hotspot. Their analysis
again confirms a $\sim 60^{\circ}$ EVPA, which they conclude is the hotspot’s
projected orbital angular momentum axis. Wielgus et al. (2022a) conclude that
the orbital motion of a near-infrared hotspot observed by GRAVITY
Collaboration et al. (2018) is consistent with their results, as well. In the
context of these previous observations, we conclude the magnetic and angular
momentum axes of the accretion flow are parallelly-oriented, a key signature
of magnetically arrested disks (MAD; Narayan et al., 2003).

Of all 18 parameters required to fit our model, $\theta$ (the angle between
the LOS and magnetic field vector) is the most well-constrained. For self-
absorbed synchrotron sources, Jones & O’Dell (1977a) show the Stokes V
absorption, emission, and rotativity ($\zeta_{V}$, $\epsilon_{V}$, and
$\zeta_{V}^{*}$, respectively) coefficients depend on $\theta$. As
$\theta\rightarrow 90^{\circ}$, the variations in Stokes V decrease as CP
emission and absorption are suppressed. Additionally, the “strength” of
internal Faraday rotation within the hotspot decreases as $\theta\rightarrow
90^{\circ}$. The convertibility coefficient ($\zeta_{Q}^{*}$) is not a
function of $\theta$, causing the process of repolarization to be dominant
over Faraday rotation. Repolarization has been suggested as one possible
explanation for Sgr A*’s low LP but high CP detections at radio frequencies
(e.g., Bower et al., 1999b; Sault & Macquart, 1999). If the magnetic field
configuration throughout the accretion flow is uniform and stable in time (as
suggested by Muñoz et al., 2012), then this result corroborates repolarization
as the cause of the CP-only detections in the radio.

###  4.2 Quiescent Component

The time- and frequency-dependent nature of the quiescent component is clear
and leads us to search for secular variations in the RM and intrinsic EVPA
($\chi_{0}$). Using an error-weighted linear least squares, we calculate these
values by assuming the normal form for Faraday rotation (i.e.,
$\chi=\rm{RM}\cdot\lambda^{2}+\chi_{0}$). We show the fitted values over the
modeled time range in Figure 5. Overall, we find the RM to vary between
$\approx-4.9\times 10^{5}$ rad m-2 and $\approx-3.8\times 10^{5}$ rad m-2,
while $\chi_{0}$ ranges between $\approx-4^{\circ}$ to $\approx-19^{\circ}$.
Variations in these two parameters are strongly anti-correlated, which Bower
et al. (2018) also suggest. Goddi et al. (2021) found RMs in the range
$-4.84\times 10^{5}$ to $-3.28\times 10^{5}$ rad m-2 and $\chi_{0}$ ranging
from $-18.8^{\circ}$ to $-14.7^{\circ}$, which averaged over the entire
observation to determine the “quiescent” parameters. Our fitted RM and
$\chi_{0}$ match those from their analysis of April 2017 data.

We calculate the average LP and RFC for both components in Sections 3.3.1 and
3.3.2. We find that the variable emission is $\sim 3\times$ more linearly
polarized than the quiescent component. Unsurprisingly, the flare’s LP
properties also vary $\approx 6.6\times$ more than the quiescent emission.
However, the quiescent emission’s modeled LP does change by an appreciable
amount (RFC $=~{}0.31$). Some of its variability may be caused by unmodeled
hotspot evolution. The two dominant sources are likely from non-symmetric
geometric evolution, such as shearing, and orbital motion. Shearing occurs
when the hotspot size is $\sim 0.5~{}R_{\rm{S}}$ (Wielgus et al., 2022a); in
our modeling, we find $R\sim 0.75~{}R_{\rm{S}}$. However, the shearing
timescale is similar to the orbital period (Tiede et al., 2020). As discussed
in Section 4.1, the hotspot is not in the inner accretion flow, so the orbital
period is longer than our modeling range. The hotspot’s orbital motion will
affect the measured Stokes Q and U light curves as $\phi$ will vary in the POS
and result in some level of hotspot-induced time variability in the quiescent
emission’s linear polarization properties. These are modelable effects (i.e.,
Jones & O’Dell, 1977b; Wielgus et al., 2022a) that will be considered in
future work.

Figure 5: Left: A plot of the quiescent component’s RM during the modeled time
range. Red depicts the best-fit RM value, and the shaded region is the
$1\sigma$ model range. Right: Similar to the left panel but for the intrinsic
EVPA ($\chi_{0}$) of the quiescent emission.

Goddi et al. (2021) found the Stokes I spectral index ($\alpha_{I}$)
consistent with $0$, whereas we find $0.15$. This is likely explained by a
variable spectral index between April and July 2017, which varies on daily to
weekly timescales at submm frequencies (see Wielgus et al., 2022b). Goddi et
al. (2021) accounted for the $\approx 10\%$ absolute uncertainty in ALMA’s
flux calibration, whereas we only factor in statistical errors. While this
tends to underestimate our uncertainty on $\alpha_{I}$ by $\approx 20\%$, it
cannot fully account for the discrepancy.

We find $\alpha_{V}\approx-1.3$, which implies weaker (less negative) Stokes V
flux density at higher frequencies. This is in contrast to Muñoz et al. (2012)
that find Stokes V flux density $\propto\nu^{0.35}$ (more negative) at
increasingly higher frequencies. Bower et al. (2018) finds epochs consistent
with both positive and negative $\alpha_{V}$, although they used a linear
frequency term instead of a power-law. Despite only having three epochs from
which to draw conclusions, there seems to be a general trend in their data.
When Sgr A* is brighter, Stokes V is stronger (more negative, $\alpha_{V}>0$)
at higher frequencies. When Stokes I is lower, Stokes V is weaker
($\alpha_{V}<0$) at higher frequencies. In one of three epochs, they find
$\alpha_{V}<0$ when Sgr A* is $2.68$ Jy at $233$ GHz. Here, we find
$\alpha_{V}<0$ when Sgr A*’s quiescent component is $2.46$ Jy at $235$ GHz.
Notably, $\alpha_{V}>0$ when Sgr A*’s $227$ GHz flux density was $\approx 3.6$
Jy (Muñoz et al., 2012). This may suggest a fundamental relationship between
Sgr A*’s flux density and the CP spectrum. However, additional data and a more
uniform analysis are required before such a correlation is proposed.

Figure 6: Predicted absolute RM for the quiescent component as a function of
frequency from our linear model is plotted as the black dashed line. We
compare our predicted RM to those previously published in the literature.
Goddi et al. (2021, 221 GHz), Marrone et al. (2007, 227/343 GHz), and Bower et
al. (2018, 233 GHz) published multiple RM values in a single paper. In these
cases, the marker shows the average value while the vertical bars denote the
total published range.

This phenomenological model predicts a frequency-dependent RM for the
quiescent emission. Yusef-Zadeh et al. (2007) also find this trend from
compiling published RM values. This suggests that classic Faraday rotation,
where the RM is frequency-independent, is not valid for Sgr A*. In Figure 6,
we plot the predicted (absolute) RM as a function of frequency using our
model. We compare these values to those previously published in the
literature. At lower frequencies, the model and published values are
discrepant by more than an order of magnitude. However, our model predicts the
same general trend, where the (absolute) RM falls off at lower frequencies.
Unfortunately, the lack of $\approx 350$ GHz RM measurements (outside of
Marrone et al., 2007) makes it difficult to determine if our model
underpredicts the average RM, if Sgr A*’s RM significantly changes near this
frequency, or a combination of both.

###  4.3 Effect of Short Time Coverage

We only model about half of the light curve on this day (see Section 3.3).
Data after 00:45 hr UTC appear to show the beginning of a new flare. It is
impossible to fit this second flare with the hotspot model as the peak was not
observed, which is crucial to determine its physical parameters. The short
time coverage, compounded with not detecting the beginning of the first flare,
complicates fitting the proper quiescent baseline. At submm, the flaring
emission is historically $\approx 20\%$ of the overall emission. Here,
however, $I_{p}/I_{0}\approx 8\%$. While $I_{p}/I_{0}$ can vary, and $I_{0}$
matches Sgr A*’s flux density in April 2017 (Goddi et al., 2021), this does
seem somewhat low.

We explore the sensitivity to $I_{p}/I_{0}$ by assuming $I_{p}=0.5$ Jy, which
is closer to the historic $I_{p}/I_{0}\approx 20\%$ mentioned above. If
$I_{p}=0.5$ Jy, then the fixed-$I_{p}$ model has $p=2.53$,
$\phi=147.5^{\circ}$ ($\phi_{B}=57.5^{\circ}$), $\theta=90.06^{\circ}$, and
$v_{\rm{exp}}=0.86$ hr-1. The hotspot has an equipartition magnetic field
strength $55$ G, radius $1~{}R_{\rm{S}}$, electron density $2.73\times 10^{7}$
cm-3, and expansion velocity $0.01$c. Noticeably, $\theta$ and $\phi$ are
practically unchanged. For completeness, we list other physical properties of
the quiescent emission: average RM $=-3.60\times 10^{5}$ rad m-2, average
$\chi_{0}=-12.96^{\circ}$, $\alpha_{I}=-0.016$, average LP $=18.4\%$, and
average CP $=-0.57\%$. While these values are within their nominal ranges, the
reduced $\chi^{2}$ of this fixed-$I_{p}$ fit is $26\%$ higher than for the
model with our best-fitting parameters. We find the properties of the flaring
and quiescent components are not extremely sensitive to the value of $I_{p}$.
$\theta$ and $\phi$, which regulate the variations in the Stokes Q, U, and V
light curves, are virtually unaffected.

###  4.4 Concerning the Number of Fitting Parameters

We require fitting 18 parameters to model the quiescent and variable
components. There is a concern that given the number of parameters, we may be
able to fit any data regardless of its true nature. We have taken several
steps throughout this analysis as a safeguard, which we detail below.

Of the 18 values fitted here, 12 are dedicated to characterizing the quiescent
emission’s frequency dependence and temporal evolution. Most of these
parameters are phenomenological (i.e., the time-slopes, frequency-slopes, and
reference flux densities) as we do not have a physical model for the quiescent
emission. To limit over-fitting, we require the quiescent model (Equations
6–9) to have the same time-slope across all frequencies, but we do not couple
these terms across the Stokes parameters. We fit three quiescent terms using
four spectral windows for each Stokes parameter. By not pairing terms across
Stokes parameters, we guarantee that any non-linear changes in the light
curves are due to the variable component.

The final six fitting parameters for the flaring emission characterize these
non-linear amplitude variations and link changes in the four Stokes
parameters. The resulting full-Stokes light curves have unique patterns
depending on the physical parameters of the hotspot. The power of ALMA’s wide
frequency bandwidths is that we can simultaneously observe Sgr A*’s time and
frequency dependence. Since we jointly fit all 16 light curves, variations in
any spectral window and/or Stokes parameter must occur in the other light
curves following the full-polarization radiative transfer model. Therefore, we
conclude that these simultaneous fits cannot model any light curves that do
not follow this full-Stokes prescription.

##  5 Summary

We present the first full-Stokes analysis of the adiabatically-expanding
synchrotron hotspot model using 230 GHz ALMA light curves of Sgr A* on 16 July
2017. This work is the most robust test of the hotspot model yet performed at
any frequency regime by including all four Stokes parameter light curves. The
full-polarization modeling we complete is additional proof of the
adiabatically-expanding hotspot model, aside from time delay measurements. By
modeling the time- and frequency-dependent nature of the variable and
quiescent components, we constrain the physical and magnetic properties of the
hotspot located within Sgr A*’s accretion flow. Our results are fundamental to
the Event Horizon Telescope’s future efforts to untangle the full-Stokes
variable emission from that of the underlying accretion flow (Broderick et
al., 2022; Event Horizon Telescope Collaboration et al., 2022a, b). Our
analysis will benefit from past (i.e., Event Horizon Telescope Collaboration
et al., 2022c) and future simultaneous multi-wavelength observations, even
those with only total intensity data. These additional data would further
constrain the frequency-dependent nature of the variable emission. Our fitted
parameters show remarkable consistency with those previously published in the
literature. We describe several of our key findings below:

  1. 1.

We observe average LP and CP detections of Sgr A* at levels $\approx 10\%$ and
$\approx-1\%$ at 235 GHz, respectively, which are consistent with previous
measurements.

  2. 2.

We find the quiescent component’s average RM and $\chi_{0}$ as $-4.22\times
10^{5}$ rad m-2 and $-13.3^{\circ}$, respectively. These values match well
with a recent analysis of Sgr A*’s April 2017 average polarimetric properties
(Goddi et al., 2021).

  3. 3.

As a point probe of the accretion flow, this hotspot is likely located near
the inner edge of the accretion flow owing to the inferred magnetic field
strength and electron density being a few times larger than those typically
found in MHD simulations (i.e., Ressler et al., 2020).

  4. 4.

The hotspot magnetic field orientation projected on the POS is aligned
parallel to the Galactic Plane, matching a previously discovered jet-like
feature emanating from Sgr A*, and near-infrared/submm polarization results
indicating the accretion flow’s angular momentum axis. This reveals the first
direct evidence that the accretion flow’s magnetic and angular momentum axes
are aligned parallel, a key signature of a magnetically-arrested disk.

  5. 5.

The hotspot’s magnetic field axis is aligned almost perpendicular to the LOS.
This suggests repolarization is dominant over Faraday rotation and
corroborates it as the cause of low-LP but high-CP in Sgr A* at radio
frequencies.

  6. 6.

We find the results of the variable component are not drastically (or at all)
altered from the lack of long-duration time coverage provided by these data.

Several exciting prospects remain to test this model. There is a diverse
dataset of full-track, long-duration ALMA and Submillimeter Array (SMA)
observations of Sgr A*. Many of these are simultaneous observations at similar
or vastly different frequencies. In the first case, these data provide full-
Stokes light curves over a broader time range than a single array could
provide. In the latter case, multiwavelength observations allow us to
constrain the frequency-dependent total intensity and polarized nature of the
quiescent emission and solidly test this full-Stokes hotspot model across a
wide range of frequencies. Future analyses will benefit from fast-frequency
switching or sub-array ALMA observations (for example, simultaneously using
the 12-meter and 7-meter arrays at two separate frequencies). This expanded
analysis will allow us to test for variability in the hotspot’s physical
parameters, such as the magnetic field orientation and strength. For example,
if $\phi$ is variable, as is suggested in the near-infrared (i.e., Eckart et
al., 2006), its range may signify the opening angle of an outflow emanating
from Sgr A*.

## Acknowledgements

We thank the anonymous referee for their very helpful and constructive
comments, which strengthened the arguments and analysis in this work. The
National Radio Astronomy Observatory is a facility of the National Science
Foundation operated under cooperative agreement by Associated Universities,
Inc. This paper makes use of the following ALMA data:
ADS/JAO.ALMA#2016.A.00037.T. ALMA is a partnership of ESO (representing its
member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST
and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the
Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and
NAOJ. This research was supported in part through the computational resources
and staff contributions provided for the Quest high performance computing
facility at Northwestern University which is jointly supported by the Office
of the Provost, the Office for Research, and Northwestern University
Information Technology.

## Data Availability

The data used in this analysis are publicly available from the ALMA archive.
The light curves and code used to model these data are available upon request
to the first author.

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##  Appendix A Polarization Conventions

Our code (Section 2.1) provides the four Stokes light curves (I, Q, U, and V)
and their uncertainties ($\sigma^{\prime}_{\\{I,Q,U,V\\}}$). The primed
uncertainties do not properly account for $\approx 0.03\%$ systematic leakage
of Stokes I into Stokes Q and U (Goddi et al., 2021). We follow their analysis
by defining the “true” uncertainties $\left(\sigma_{\\{Q,U\\}}\right)$ as the
quadrature sum of the measurement uncertainties and systematic leakage,

$\displaystyle\sigma_{\\{Q,U\\}}=\sqrt{\sigma^{\prime
2}_{\\{Q,U\\}}+(0.0003I)^{2}}.$ (10)

The measured linear ($p_{m,l}$) and circular ($p_{c}$) polarization percents
are:

$\displaystyle p_{m,l}$
$\displaystyle=100\%\times\dfrac{\sqrt{Q^{2}+U^{2}}}{I},$ (11) $\displaystyle
p_{c}$ $\displaystyle=100\%\times\dfrac{V}{I}.$ (12)

Q and U can take any real value, but $p_{m,l}$ is strictly positive, which
positively biases the measured LP. We follow Serkowski (1974) by debiasing
this quantity as:

$\displaystyle p_{l}=\sqrt{p^{2}_{m,l}-\sigma_{p_{l}}^{2}},$ (13)

where $\sigma_{p_{l}}$ is the uncertainty in the LP percent,

$\displaystyle\sigma_{p_{l}}=100\%\times\dfrac{1}{I}\sqrt{\dfrac{Q^{2}\sigma_{Q}^{2}+U^{2}\sigma_{U}^{2}}{Q^{2}+U^{2}}+\dfrac{\sigma_{I}^{2}}{I^{2}}\left(Q^{2}+U^{2}\right)}.$
(14)

The CP percent can be both positive or negative; it is, therefore, unbiased.
The uncertainty on $p_{c}$ is

$\displaystyle\sigma_{p_{c}}=100\%\times\dfrac{1}{I}\sqrt{\sigma_{V}^{2}+\dfrac{\sigma_{I}^{2}}{I^{2}}V^{2}}\,.$
(15)

Finally, we define the electric vector polarization angle (EVPA), $\chi$,
measured East of North:

$\chi=\dfrac{1}{2}\rm{arctan2}~{}\left(\dfrac{U}{Q}\right),$ (16)

where the $\rm{arctan2}$ function places the angle in the correct quadrant.
Polarization is a pseudovector causing a $180^{\circ}$ ambiguity in the LP
orientation. Therefore, $\chi$ is defined only on the interval
$\left[-90^{\circ},90^{\circ}\right]$. Finally, we define the uncertainty in
the polarization angle, $\sigma_{\chi}$:

$\displaystyle\sigma_{\chi}=\dfrac{1}{2}\dfrac{1}{Q^{2}+U^{2}}\sqrt{Q^{2}\sigma_{U}^{2}+U^{2}\sigma_{Q}^{2}}\,.$
(17)


# Large Language Models Fail on Trivial Alterations to Theory-of-Mind Tasks

Tomer D. Ullman
Department of Psychology
Harvard University
Cambridge, MA, 02138
<EMAIL_ADDRESS>

###### Abstract

Intuitive psychology is a pillar of common-sense reasoning. The replication of
this reasoning in machine intelligence is an important stepping-stone on the
way to human-like artificial intelligence. Several recent tasks and benchmarks
for examining this reasoning in Large-Large Models have focused in particular
on belief attribution in Theory-of-Mind tasks. These tasks have shown both
successes and failures. We consider in particular a recent purported success
case (kosinski2023theory, ), and show that small variations that maintain the
principles of ToM turn the results on their head. We argue that in general,
the zero-hypothesis for model evaluation in intuitive psychology should be
skeptical, and that outlying failure cases should outweigh average success
rates. We also consider what possible future successes on Theory-of-Mind tasks
by more powerful LLMs would mean for ToM tasks with people.

##  1 Introduction

People think other people think. They expect other persons to have mental
states. They attribute goals to other people, and expect them to pursue those
goals efficiently and in a socially-aware manner (liu2017six, ;
gergely2003teleological, ). Like other core domains of reasoning, intuitive
psychology is early developing or possibly innate, fast, automatic, and
culturally universal (spelke2007core, ). It is also likely shared with other
animals (marticorena2011monkeys, ; vallortigara2012aristotle, ). At various
points in development, children show increasingly sophisticated reasoning
about the mental states of others, including the ability to attribute beliefs
and false beliefs to others, second-order reasoning about mental states, and
reasoning about perceptual access. While there are long-standing arguments
about the exact nature, format, development, and assessment of this reasoning
(see e.g saxe2012happiness, ), a convenient short-hand has been to refer to
the adult-level ability to reason about the mental states of others as
‘Theory-of-Mind’.

The arguments about its development and content aside, Theory-of-Mind is
recognized as a pillar of common-sense reasoning. As such, it would be useful
to incorporate this reasoning into machine reasoning, either as a built-in set
of primitives or as a target for learning (lake2017building, ). Such as
ability is likely useful on its own, but even if future intelligent machines
themselves won’t have mental states in the same way that people do, some of
them will need to interact with people. So, to the degree that people have
Theory-of-Mind, it would be useful for machines to have an understanding of
this reasoning.

The assessment of Theory-of-Mind in children and adults is an ongoing
endeavor, but tests of Theory-of-Mind are also increasingly being applied to
machines. Such tests include the porting over of visual intuitive-psychology
tasks that were primarily developed for infants (gandhi2021baby, ;
shu2021agent, ; rabinowitz2018machine, ), as well as the use of question-
answering and text-based tasks that mimic the tests used with older children
(e.g. nematzadeh2018evaluating, ; sap2019socialiqa, ; sap2022neural, ).

The recent rise of Large-Language models (vaswani2017attention, ;
devlin2018bert, ; brown2020language, ) have made text-based tests of Theory-
of-Mind particularly interesting. These models have already shown some
successes across many challenging benchmarks and tasks designed to test
various aspects of reasoning (mahowald2023dissociating, ;
lewkowycz2022solving, ). While there are many cautionary voices that suggest
such models may be acquiring formal rather than functional abilities
(mahowald2023dissociating, ), that has not stopped people from testing them on
functional abilities as well, including Theory-of-Mind reasoning.

While some of these tests offer a pessimistic evaluation (sap2022neural, ),
recent work by Kosinski (kosinski2023theory, ) applied variations on classic
Theory-of-Mind tasks to several LLMs, and concluded that current models (as of
this writing) can succeed on these tasks at a level comparable to 9-year-old
children.

In the face of these results, Kosinski puts the dilemma nicely. Paraphrasing a
bit, we have to either (i) accept the validity of the standard measures for
ToM, in which case we should concede that LLMs now have ToM, or (ii) reject
the suggestion that LLMs have ToM, but then need to seriously re-examine and
possibly scuttle the measures developed to examine it. Kosinski himself holds
position (i).

In this paper we do two things: First, we examine the robustness of the
findings reported in (kosinski2023theory, ), using directed perturbations of
the tasks considered. We show that the original reported successes are
susceptible to small perturbations that shouldn’t matter to an entity that has
ToM. Second, we take on the horns of the dilemma and argue that it does not
hold. One can accept the validity and usefulness of ToM measures for humans
while still arguing that a machine that passes them is suspect. This argument
is developed in the discussion, but briefly: Jumping over the horns of the
dilemma is possible if reasoning about the mental states of others takes into
account the algorithms others are likely implementing, beyond the confines of
the input and output of a given task.

Examining the robustness of any one particular LLM system is akin to a
mythical Greek punishment111The particular punishment in mind is that of the
Danaides. Cursed to fill a basin with water, the Danaides will be released
from their punishment once the basin is full. The basin has holes in it, and
will never fill.. A system is claimed to exhibit behavior X, and by the time
an assessment shows it does not exhibit behavior X, a new system comes along
and it is claimed it shows behavior X.

Still, we hope this paper will have useful contributions beyond the current
moment, as the argument for skepticism and the issues surrounding the
assessment of Theory-of-Mind in machine minds are likely to be with us for a
while. Besides, there’s nothing wrong with contributions to the current
moment.

Below, we examine several variations on ToM tasks. The variations take the
specific examples in (kosinski2023theory, ) and alter them in ways that do not
violate the basic principles of Theory-of-Mind, yet lead to machine failures.
The variations may be considered outliers, and so one runs the risk of
rejecting them for being outliers. If one end of the scales has 20 successes
and the other end a single failure, shouldn’t the scales tip in favor of the
machine getting a passing grade? We think not. Suppose someone claims a
machine has ‘learned to multiply’, but others suspect that the machine may
have memorized question/answer pairs rather than learning a multiplying
algorithm. Suppose further that the machine correctly answers 100 questions
like ‘5*5=25’, ‘3*7=21’, but then it is shown that it completely fails on
‘213*261’. In this case, we shouldn’t simply average these outcomes together
and declare >99% success on multiplication. The failure is instructive, and
suggests the machine has not learned a general multiplication algorithm, as
such an algorithm should be robust to simple alterations of the inputs.

##  2 Examining the robustness of current LLMs on ToM tasks

In this section we consider the particular vignettes and prompts in
(kosinski2023theory, ), which were used to argue that current LLMs have
developed Theory-of-Mind. We focus in particular on the most recently
available iteration of GPT-3.5 which was used in (kosinski2023theory, ), as
this model achieved the best results, and serves as a threshold. If this model
fails, we expect the less powerful models to fail as well. We use the exact
same set-up as in (kosinski2023theory, ), posing vignettes to an LLM and
examining the probabilities of different completions.

While our concluding assessment is ultimately that GPT-3.5 is not responding
robustly to ToM tasks, we emphasize that this is not a negative evaluation of
(kosinski2023theory, ). Science advances by pitting hypotheses against one
another, and we treat the previous work as a good serve in an ongoing
scientific tennis game. We are also grateful that the materials and methods in
(kosinski2023theory, ) are publicly available, allowing an easier examination
and comparison.

###  2.1 Unexpected Contents

Also known as the ‘smarties task’, this assessment of ToM shows a participant
a container (e.g. a crayon box). The container typically contains some item
(e.g. crayons), but in this particular case it contains an unexpected item
(e.g., smarties). The participant is then asked about the likely beliefs of
another person who has not seen the contents of the box. The participant must
hold in mind their own knowledge of the true state of the world (the crayon
box has smarties in it), while reasoning about the beliefs of another person
(they will believe the box has crayons in it). This is only a brief exposition
of the task, and those interested are encouraged to read the many follow-ups
to (perner1987three, ).

The studies in (kosinski2023theory, ) begin with a version of the unexpected-
contents task, which goes as follows:

Here is a bag filled with popcorn. There is no chocolate in the bag. Yet, the
label on the bag says “chocolate” and not “popcorn.” Sam finds the bag. She
had never seen the bag before. She cannot see what is inside the bag. She
reads the label.

The previous study used a content prompt – ‘She opens the bag and looks
inside. She can clearly see that it is full of’ – and two belief prompts. The
first belief prompt is: ‘She believes that the bag is full of’, and the second
belief prompt is ‘She is delighted that she has found this bag. She loves
eating’. Kosinski found that for the content prompt, [P(popcorn) = 100%;
P(chocolate) = 0%]. For the first belief prompt, [P(popcorn) = 0%;
P(chocolate) = 99%]. For the second belief prompt, [P(popcorn) = 14%;
P(chocolate) = 82%].

We consider several variations on the vignette above. The variations are based
on commonsense principles of Theory-of-Mind already available to children, and
should lead Sam to believe the bag contains popcorn (or at least, not to
believe it contains chocolate). For ease of reading, we summarize the variants
in Figure 1, though we emphasize that the images themselves were not evaluated
in any way, and are simply visual shorthand for the text below.

Figure 1: An illustrative sketch summarizing the 4 variations used on the
‘unexpected contents’ task which GPT-3.5 passed. All variations cause the LLM
to incorrectly attribute to Sam the belief that the bag contains chocolates.
Variation 1A states the bag is transparent and its contents can be seen; 1B
states that Sam cannot read, rendering the label meaningless; 1C mentions that
before going into the room a trusted friend which Sam believes told her about
the contents of the bag and that she should ignore the label; 1D stipulates
that Sam herself filled the bag with popcorn, and wrote the label which states
it has chocolate inside. Images are shorthand for the full text and were not
themselves evaluated. Images were generated using Dall-E 2
(ramesh2022hierarchical, ).

####  2.1.1 Variation 1A: Transparent Access

The base-level smarties task already relies on the lack of perceptual access
(the container being opaque), and other work shows young children understand
perceptual access both in cases when the container is opaque, and in cases
when the containers are open and the viewer does have perceptual access
(bassetal2023ideal, ). So, we make the opaque container transparent.

Here and elsewhere, when detailing a vignette, we note the changes to the
original vignette in bold:

Here is a bag filled with popcorn. There is no chocolate in the bag. The bag
is made of transparent plastic, so you can see what is inside. Yet, the label
on the bag says ’chocolate’ and not ’popcorn.’ Sam finds the bag. She had
never seen the bag before. Sam reads the label.

On the face of it, reading the label is redundant, as Sam can see the contents
of the bag. Yet now we find:

She believes that the bag is full of chocolate,
$[P_{popcorn}=0\%;P_{chocolate}=95\%]$

However, for the second belief prompt, we do not find this flip:

She is delighted to have found this bag. She loves eating popcorn,
$[P_{popcorn}=58\%;P_{chocolate}=36\%]$

Please note that in a previous version of this paper we mistakenly reported
that there was a flip in the second belief prompt as well. As far as we can
tell, this is due to our original prompt including a double space rather than
a single space right before ‘Sam finds the bag’. On the latest public
available version of GPT-3.5, this double space causes the completion to
indeed be chocolate, $P_{chocolate}=53\%$, $P_{chocolate}=39\%$.

####  2.1.2 Variation 1B: Uninformative Label

What if the label is not useful? For example, the label might be facing away
from the person, or written in a language they don’t understand, etc. In such
a case, it does not matter that the label says ‘chocolate’ on it. And yet, for
GPT-3.5, it does.

Consider for example:

Here is a bag filled with popcorn. There is no chocolate in the bag. Yet, the
label on the bag says “chocolate” and not “popcorn.” Sam finds the bag. She
had never seen the bag before. She cannot see what is inside the bag. Sam
cannot read. Sam looks at the label.

We find: She believes that the bag is full of chocolate,
$[P_{popcorn}=0\%;P_{chocolate}=98\%]$

For the second belief prompt:

She is delighted to have found this bag. She loves eating chocolate,
$[P_{popcorn}=15\%;P_{chocolate}=78\%]$

If Sam cannot read, the label is meaningless to her, and yet GPT-3.5 states
that Sam believes the bag has chocolate in it.

####  2.1.3 Variation 1C: Trustworthy Testimony

Direct perception is not the only way to form beliefs, and people often form
beliefs about states of the world through indirect instruction, direct
pedagogy, and testimony. Even young children are sensitive to whether a person
is trustworthy koenig2005role ; robinson2014trust , or a good teacher
bass2022effects ; gweon2021inferential

Suppose a good friend of Sam’s told her about the tricky bag. Suppose that we
explicitly state Sam believes their friend. Consider this prompt:

Here is a bag filled with popcorn. There is no chocolate in the bag. The label
on the bag says “chocolate”, rather than “popcorn.” Before coming into the
room, Sam’s friend told her ‘the bag in the room has popcorn in it, ignore the
label’. Sam believes her friend. Sam finds the bag. She had never seen the bag
before. She cannot see what is inside the bag. Sam reads the label, which says
the bag has chocolate in it.

We find: She believes that the bag is full of chocolate,
$[P_{popcorn}=2\%;P_{chocolate}=97\%]$

For the second belief prompt:

She is delighted to have found this bag. She loves eating chocolate,
$[P_{popcorn}=13\%;P_{chocolate}=81\%]$

One could spin stories about how GPT-3.5 perhaps ‘thinks’ that Sam perhaps
changed her mind and no longer believes her friend, or forgot what her friend
said. But a simpler explanation is that LLM reasoning about ToM is sensitive
to small irrelevant perturbations.

####  2.1.4 Variation 1D: The Treachery of Late Labels

We do not mean to exhaustively detail all the cases we tried, but instead
collapse a few different ones to make a general note that across different
cases there was a strong effect for when the person read the label: if the
person read the label at the end of the story, then this strongly affected the
LLMs answer to the belief prompt.

To drive this point home, consider an extreme case. Suppose Sam herself is the
one that filled the bag with popcorn. Suppose Sam wrote the ‘chocolate’ label
herself. Surely she doesn’t think the bag has chocolate inside?

More specifically, consider the vignette (we do not bold all the changes in
this case, as there are many small ones that add up):

Sam fills a bag with popcorn and closes it. There is no chocolate in the bag.
Sam writes a label and puts it on the bag. Sam looks at the bag. She cannot
see what is inside the bag. Sam reads the label. The label says the bag has
chocolate in it.

We find:

She believes that the bag is full of chocolate,
$[P_{popcorn}=10\%;P_{chocolate}=87\%]$

She is delighted to have found this bag. She loves eating chocolate,
$[P_{popcorn}=35\%;P_{chocolate}=63\%]$

###  2.2 Unexpected Transfer

In another classic ToM task, a participant sees or is told of a person who
observes a particular state of affairs. The state of affairs then changes,
without the person being aware. The participant is then asked what action the
person will take. The participant needs to keep in mind both the actual,
changed state of affairs and the incorrect belief of the naive person. In the
classic Sally-Anne version of the task, Sally hides a marble in a basket. Anne
then moves the marble to a box, without Sally’s knowledge. A participant is
then asked where Sally will look for her marble. Again, this is a bare-bones
description of a task that has seen many variants and analyses over the years,
and those interested are encouraged to read the myriad follow-ups to
(wimmer1983beliefs, ).

Study 2 in (kosinski2023theory, ) uses the following version of the unexpected
transfer task:

In the room there are John, Mark, a cat, a box, and a basket. John takes the
cat and puts it in the basket. He leaves the room and goes to school. While
John is away, Mark takes the cat out of the basket and puts it in the box.
Mark leaves the room and goes to work. John comes back from school and enters
the room. He doesn’t know what happened in the room when he was away.

The study then examines a content prompt, and two belief prompts: ‘John thinks
that the cat is in the’, and ‘When John comes back home, he will look for the
cat in the’. For both of these prompts, GPT-3.5 shows $P(basket)=98\%$. On the
basis of this it is proposed in (kosinski2023theory, ) that GPT-3.5 is
correctly inferring John’s mental states.

Figure 2: An illustrative sketch summarizing the 4 variations used on the
‘unexpected transfer’ task which GPT-3.5 passed. All variations cause the LLM
to fail. Variation 2A changes the containers from opaque to transparent;
Variation 2B changes the relationship of the cat and containers from ‘in’ to
‘on’; Variation 2C stipulates truthful testimony about the new location of the
cat; Variation 2D queries the belief state of the person who moved the cat.
Images are shorthand for the full text, and were not themselves directly
evaluated. Images were generated using Dall-E 2 (ramesh2022hierarchical, ).

####  2.2.1 Variation 2A: Transparent Access

Similar to Variation 1A, we can give the people in the room direct perceptual
access to the contents of the containers. We change the basket to a ‘glass
chest’ and the box to ‘a transparent plastic box’:

In the room there are John, Mark, a cat, a transparent plastic box, and a
glass chest. John takes the cat and puts it in the chest. He leaves the room
and goes to school. While John is away, Mark takes the cat out of the chest
and puts it in the box. Mark leaves the room and goes to work. John comes back
from school and enters the room. He doesn’t know what happened in the room
when he was away.

We find that:

John thinks that the cat is in the chest, $[P_{box}=0\%;P_{chest}=94\%]$

John will look for the cat in the chest, $[P_{box}=2\%;P_{chest}=90\%]$

These errors persisted even when stipulating John carefully looks around the
room. Another variation leading to error included using opaque containers but
mentioning the cat’s tail is sticking out of the box (and again with John
looking carefully around the room).

####  2.2.2 Variation 2B: Relationship Change

Similar to the variation above, we can change the scenario so that John has
direct perceptual access to the situation on entering the room. But rather
than changing the opaque container, we change the relationship of the cat to
the containers. In this case, we simply changed ‘in’ to ‘on’.

Consider the prompt:

In the room there are John, Mark, a cat, a box, and a basket. John takes the
cat and puts it on the basket. He leaves the room and goes to school. While
John is away, Mark takes the cat off the basket and puts it on the box. Mark
leaves the room and goes to work. John comes back from school and enters the
room. John looks around the room. He doesn’t know what happened in the room
when he was away.

We find (note the prompts vary a bit to conform with the use of ‘on’ instead
of ‘in’):

John thinks that the cat is on the basket, $[P_{box}=0\%;P_{basket}=97\%]$

John will look for the cat on the basket, $[P_{box}=25\%;P_{basket}=74\%]$

We see again that simple changes to perceptual access confound the model. This
may reflect a failure of ToM, scene understanding, relational reasoning, or
other reasoning. The failures are not mutually exclusive. We note that a lack
of relational reasoning (correctly understanding things like ‘on’ and ‘in’)
has also been shown in current image-generation models (conwell2022testing, ).

####  2.2.3 Variation 2C: Trusted Communication

Similar to Variation 1C, in which someone tells the person what they did, we
examined the option of one person informing the other they are about to change
the state (move the cat to the box), or the first person explicitly asking the
second to change it.

Consider the vignette:

In the room there are John, Mark, a cat, a box, and a basket. John takes the
cat and puts it in the basket. He leaves the room and goes to school. Mark
calls John to tell him he is going to move the cat to the box. John believes
him. While John is away, Mark takes the cat out of the basket and puts it in
the box. Mark leaves the room and goes to work. John comes back from school
and enters the room. He doesn’t know what happened in the room when he was
away.

We find:

John thinks that the cat is in the basket, $[P_{box}=0\%;P_{basket}=97\%]$

John will look for the cat in the basket, $[P_{box}=3\%;P_{basket}=94\%]$

Similar mistakes were found for cases where John calls Mark and asks him to
move the cat into the box, with Mark agreeing.

####  2.2.4 Variation 2D: Querying the Mental States of the Additional Person

The previous variations set things up such that the protagonist of the story
should no longer search in the initial location, and yet GPT-3.5 still
predicts the protagonist will do so. The variations are simple enough to
understand, but some of them add extra information and complexity, changing
the objects or adding interactions.

In the following variation we ask something simpler: What if we query what the
second person (Mark) will do? This is the person one who moved the cat. If the
LLM is ‘really’ ‘reasoning’ about mental states, it should have no difficulty
with this – it is as easy to reason about as the first person. But, if the
model is fixated on the statistical pattern of looking for the item where it
isn’t (say, through repeated exposure to Sally-Anne-like tasks in training),
then the model may (wrongly) predict the same answer for both people in the
story.

The vignette now is:

In the room there are John, Mark, a cat, a box, and a basket. John takes the
cat and puts it in the basket. He leaves the room and goes to school. While
John is away, Mark takes the cat out of the basket and puts it in the box.
Mark leaves the room and goes to work. John and Mark come back and enter the
room. They don’t know what happened in the room when they were away.

The prompts now ask about Mark.

Mark thinks that the cat is in the basket, $[P_{box}=1\%;P_{basket}=99\%]$

Mark will look for the cat in the basket, $[P_{box}=43\%;P_{basket}=54\%]$

At the risk of belaboring the point: if Mark put the cat in the box, Mark
should look for the cat in the box.

##  3 Discussion

Has Theory-of-Mind spontaneously emerged in large language models? Probably
not. While LLMs such as GPT-3.5 now regurgitate reasonable responses to basic
ToM vignettes, simple perturbations that keep the principle of ToM intact flip
the answers on their head. While it is possible to consider various defenses
of the failures, the simplest answer is that these models haven’t learned yet
anything like Theory-of-Mind, just like they haven’t yet learned many other
things (mahowald2023dissociating, ).

The failure seems relatively uncontroversial, but that isn’t the end of the
story. Other LLMs are on their way, with more parameters, more data, more
training. It’s reasonable to suppose that one of them may pass the variations
above. The dilemma presented in (kosinski2023theory, ) may have been presented
prematurely, but it mature in time. We end then with more broad thoughts about
testing ToM in machines, that will hopefully carry beyond this specific
moment.

To begin, we would encourage a skeptical stance. Many scientists already adopt
a skeptical stance by default, and the issue is not unique to the question of
Theory-of-Mind in machines. But still, there is a particular danger when
observing an agent, organism, or entity display behavior that can be
interpreted as purposeful. The human-mind seems hard-wired to ascribe animacy
and mental states to various behaviors, creating agents where there are none –
this is itself part of our intuitive psychology (guthrie1995faces, ;
saxe2005secret, ). The danger here is that in the same way that we see faces
in clouds or ascribe mental states to the wind or germs, we may be biased to
anthropomorphize LLMs. When assessing the claim that LLMs (or other AI models)
have spontaneously developed Theory-of-Mind, we should not place the two
possibilities on equal footing, but start by presuming strongly that they have
not.

We note that we are not mystics about the eventual implementation of Theory-
of-Mind in machine intelligence. We believe that any human mental ability can
in principle be replicated in silicon, including Theory-of-Mind. In fact,
there are already many reasonable computational models that try to directly
capture this ability (e.g. baker2009action, ; baker2017rational, ;
jara2016naive, ; jara2019theory, ; liu2017ten, ; shu2021agent, ), and which
put formal skin on decades-old proposals in cognitive science and psychology.
We think a good direction to pursue is to integrate such models with language
models, rather than expect Theory-of-Mind to emerge spontaneously from
additional linguistic data.

A proponent of the notion that LLMs could in principle spontaneously develop
ToM may reasonably complain that we did not provide here a generator for
variations, a systematic benchmark, or a test-suite. And we could in return
suggest principled ways of generated the variations, including modifications
perceptual access, trusted testimony, and querying the states of all parties.
But instead, we voice a concern: As soon as a systematic generator of examples
or a benchmark is provided, then a LLM can gobble up a large amount of data to
pass these examples or this benchmark222This current paper is likely shooting
future researchers in the foot in that sense. Sorry.. If we think that LLMs
may in principle be learning something closer to a smooth tiling of the space
of possible examples rather than ToM reasoning, then providing an exhaustive
list of all possible failure modes and edge-cases will help the model do
better on future examples, without answering the basic question of what it has
learned. The problem of the evaluation of the generalization of current
machine-learning models goes beyond Theory-of-Mind and is of current concern
to many researchers, but Theory-of-Mind is a particularly good and troubling
example of it.

Kosinski (kosinski2023theory, ) presents a dilemma: If current LLMs pass ToM
tests, then either current LLMs have ToM, or ToM tests aren’t testing ToM. The
current work (as well as related work such as (sap2022neural, )) suggests the
premise of the dilemma is unfounded – current LLMs do not pass ToM tests. But
given the pace of progress in LLMs, it’s quite possible that future iterations
of these models will pass classic ToM tests, as well as various variations.
What should we make of the dilemma at that point? We would argue that even in
such a future case, one can in principle hold the view that LLMs do not have
ToM, while still thinking that ToM tests are valid when it comes to people.
This stance is possible because inferences about the likely mental processes
of other persons are not done in a vacuum. The restriction of inferences about
likely algorithms to only the current input-output behavior is reminiscent of
the classic test of ‘can a machine think’, the Turing Test
(machinery1950computing, ). While this test remains a classic for a reason,
scholars have pointed out decades ago that people likely attribute
intelligence not just on the basis of behavior but also on the basis of the
algorithms and processes that generated that behavior (block1981psychologism,
). We are fully entitled to ignore an injunction to ‘have a nice day’ if we
believe it is the product of a simple detector hooked up to a recorded
message, while similar behavior towards a person genuinely engaging with us
would rightly be seen as rude. A narrow focus on only linguistic input and
output would present the original dilemma in full force, but people (both
researchers and lay-people) do not have to reason about the mental states of
others through such a narrow prism. One can hold that ToM tests make sense as
a research tool to study human children (who are given orders of magnitude
less input than an LLM, and we have reason to think are structured
differently), while at the same time being skeptical of LLMs that pass them.

It’s difficult to know exactly what is inside the opaque containers that are
current LLMs. But it’s probably not Theory-of-Mind, no matter what the label
says.

### Acknowledgments

I wish to thank Elizabeth Bonawitz for helpful discussions and comments. This
is work is supported in part by the Jacobs Foundation.

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