# Assessing the impact of instrument noise and astrophysical fluctuations on
measurements of the first black hole photon ring

Alejandro Cárdenas-Avendaño  0000-0001-9528-1826 cardenas-
<EMAIL_ADDRESS>Princeton Gravity Initiative, Princeton University,
Princeton, New Jersey 08544, USA  Department of Physics, Princeton University,
Princeton, New Jersey 08544, USA  Lennox Keeble  0009-0009-5796-631X
Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
Alexandru Lupsasca  0000-0002-1559-6965 Department of Physics & Astronomy,
Vanderbilt University, Nashville, Tennessee 37212, USA

###### Abstract

Currently envisioned extensions of the Event Horizon Telescope to space will
soon target the black hole photon ring: a narrow ring-shaped imprint of a
black hole’s strong gravity produced in its images by highly bent photon
trajectories. In principle, the shape of the photon ring encodes information
about the geometry of the underlying black hole spacetime. In practice,
however, whether or not this information can be extracted from the ring shape
depends on several factors, ranging from the astrophysical details of the
emitting source (such as the magnitude of its plasma fluctuations) to the
specific configuration of the interferometric array (such as the separation
between its telescopes, or the level of noise in its instruments). Here, we
employ a phenomenological model to assess the impact of astrophysical
fluctuations and instrument noise on the inferred shape of the photon ring.
Our systematic study of several astrophysical profiles suggests that this
shape can be measured even in the presence of instrument noise across a wide
range of baselines. The measurement accuracy and precision appear relatively
insensitive to the noise level, up to a sharp threshold beyond which any
measurement becomes incredibly challenging (at least without recourse to more
sophisticated data analysis methods). Encouragingly, we find that only a few
snapshot images are generally needed to overcome the impact of astrophysical
fluctuations and correctly infer the ring diameter. Inference becomes more
challenging when analyzing the visibility amplitude in a baseline window that
is not entirely dominated by a single photon ring. Nevertheless, in most
cases, it is still possible to fit a ring shape with the correct fractional
asymmetry. These results provide excellent prospects for future precision
measurements of black hole spin and fundamental astrophysics via black hole
imaging.

##  I Introduction

Although the individual pulses of a pulsar can display significant
variability, their average is remarkably stable. Pulsar timing array
experiments rely on this stability to test a wide range of relativistic
predictions Verbiest  _et al._ (2021); Agazie  _et al._ (2023). While images
of a supermassive black hole can likewise exhibit large variability, their
time average is also expected to be highly stable: individual snapshots may
display plasma flares or other prominent transient features, but such source
fluctuations ought to wash out in the long run, leaving an image dominated by
the signature of the black hole—its photon ring Johnson  _et al._ (2020);
Gralla  _et al._ (2020). This ring is a narrow image feature produced by
photons on highly bent trajectories that orbited the black hole multiple times
on their way from source to observer. It consists of multiple subrings, each
of which shows a lensed image of the main emission, indexed by the number $n$
of photon half-orbits executed around the black hole Gralla  _et al._ (2019);
Gralla and Lupsasca (2020a). These subrings encircle the central brightness
deficit caused by the event horizon and form the part of the image that
belongs to the black hole itself, rather than its surrounding emission: the
photon ring is a persistent “stamp” on the image of the black hole that
carries information about its strong gravity, encoded in the shape of the
subrings. Planning is now underway for a space mission capable of performing
interferometric observations on baselines long enough to resolve the first
($n=1$) rings of the black holes M87* and Sgr A* Gurvits  _et al._ (2022);
Kurczynski  _et al._ (2022); Kudriashov  _et al._ (2021).

As this experimental effort unfolds, ongoing theoretical work Johnson  _et
al._ (2020); Gralla (2020); Gralla and Lupsasca (2020b); Gralla  _et al._
(2020); Paugnat  _et al._ (2022); Vincent  _et al._ (2022); Cárdenas-Avendaño
and Lupsasca (2023) has focused on deriving predictions for the
interferometric signature of the photon ring. By now, multiple studies have
investigated how feasible it might be to extract this signature from time-
averaged black hole images. Initially, the focus was on the $n=2$ photon ring,
which produces a very clean signal Gralla  _et al._ (2020); Paugnat  _et al._
(2022); Vincent  _et al._ (2022), but it has recently shifted to the $n=1$
ring Cárdenas-Avendaño and Lupsasca (2023), as it is more easily accessible
observationally and will hence be resolved first.

Deriving a sharp, general-relativistic prediction for the interferometric
signature of the $n=1$ ring is nontrivial. The reason is that, strictly
speaking, general relativity only completely fixes the shape of the limiting
$n\to\infty$ ring: an infinitely thin ring known as the “critical curve”
$\tilde{\mathcal{C}}$ Bardeen (1973). Although its angle-dependent diameter
$\tilde{d}_{\varphi}$ follows an analytically known form Gralla and Lupsasca
(2020b), this mathematical shape is not in itself observable. What is in
principle observable are the finite-$n$ rings, which converge exponentially
fast in $n$ to $\tilde{\mathcal{C}}$, but can nonetheless display corrections
in $1/n$ that remain significant at $n=1$.

In particular, the first $n=1$ photon ring has a larger width-to-diameter
ratio than subsequent rings—of order $\sim\\!10\%$—and therefore lacks a
sharply defined diameter in the image domain. Nevertheless, it _does_ admit a
sharp interferometric diameter $d_{\varphi}^{(1)}$, which can be defined in
the frequency domain by the periodic ringing that the ring produces in the
radio visibility Cárdenas-Avendaño and Lupsasca (2023). In summary, even
though general relativity (GR) only directly determines a functional form for
the angle-dependent diameter $\tilde{d}_{\varphi}$ of $\tilde{\mathcal{C}}$,
this very same functional form also constrains the interferometric ring
diameter $d_{\varphi}^{(1)}$ of the first photon ring.

Although these theoretical findings are encouraging, in practice, photon ring
measurements face several hurdles. The main challenge arises from the very
narrow width of the rings: at our present observing frequencies, resolving a
photon ring requires very-long-baseline interferometric (VLBI) observations on
baselines extending from ground stations all the way to a radio antenna in
space Johnson  _et al._ (2020); Tiede  _et al._ (2022); Lockhart and Gralla
(2022).

Besides the considerable technological advances needed to carry out such
observations in space, the narrow width of the photon ring also poses
challenges to our existing modeling approaches: to adequately resolve it, one
has to ray trace general-relativistic magnetohydrodynamic (GRMHD) simulations
of the accretion flow onto the black hole at very high image resolutions (for
more details on these types of simulations, see, e.g., Refs. Font (2008);
Porth  _et al._ (2019); Wong  _et al._ (2022); Mizuno (2022)).

Such time-averaged GRMHD simulations have shown that photon rings are
persistent, sharp features that come to dominate VLBI observations after
averaging them over sufficiently long timescales Johnson  _et al._ (2020).
Because of their onerous computational cost, however, these simulations cannot
explore the vast space of plausible configurations, which is parameterized by
many variables, including: black hole spin, observer inclination, the plasma
heating model, or accretion flow magnetization states, among many others.

Properly accounting for the time variability of sources presents another
challenge: for instance, the variability observed in Sgr A*, the supermassive
black hole at the center of our galaxy, is a factor of two smaller than the
source-integrated variability predicted by our GRMHD models Akiyama  _et al._
(2022); Wielgus  _et al._ (2022). In other words, we expect photon rings to
appear in time-averaged images, but the exact details of how this averaging is
to be carried out remain unclear.

For these reasons, prior studies of the photon ring and its interferometric
signature have mostly revolved around time-averaged images Gralla  _et al._
(2019); Johnson  _et al._ (2020); Gralla  _et al._ (2020); Paugnat  _et al._
(2022); Vincent  _et al._ (2022); Cárdenas-Avendaño and Lupsasca (2023). To
forecast future space-VLBI experiments, however, one must tackle the time
variability head-on, and do so at high resolution. In addition, accurate
forecasting must take into account the fine details of the interferometric
array: in particular, one must choose an orbit for the space telescope, and
model the physical and operational constraints imposed by the spacecraft
design and mission architecture Hudson  _et al._ (2023). If, for example, the
space dish observes with the frequencies of existing ground stations (namely,
$86$, $230$, or $345$ GHz), then its orbit determines the baseline coverage
(i.e., the subset of the frequency domain) accessible to the array.

Here, we begin to incorporate these complications into our forecasting, using
a phenomenological model built into the Adaptive Analytical Ray Tracing code
AART Cárdenas-Avendaño  _et al._ (2023). This code exploits the integrability
of light propagation in the Kerr spacetime to produce high-resolution black
hole images and their corresponding visibilities, as would be observed on long
space-to-ground baselines. We conduct a survey of parameter space to determine
how accurately the shape of the first photon ring can be inferred in the
presence of various levels of instrument noise and source variability. The
survey also varies baseline lengths and ranges as a proxy for the yet-to-be-
pinned-down orbit.

To model instrument noise, we add complex Gaussian noise to the raw, simulated
radio visibility. Meanwhile, we model astrophysical fluctuations using inoisy
Lee and Gammie (2021), a code capable of replicating many features of
accretion processes by generating realizations of Gaussian random fields
(GRFs) with tuneable spatiotemporal correlations. That is, rather than
strictly adhering to a physics-based simulation, such as in GRMHD, we instead
resort to a stochastical model of fluctuations that allows us to vary their
power and correlation structure systematically. This enables us to
parameterize our uncertainty about the scale of the fluctuations, and to
thereby offer a broad understanding of the various factors that will influence
future observations. Since the amount of time-averaging required for a future
space-VLBI mission to isolate the photon ring ultimately depends on the
variability of the underlying plasma, we consider multiple astrophysical
models with different variability and emission geometries.

The results of our analysis suggest that, even in the presence of substantial
levels of _both_ instrument noise and astrophysical fluctuations, it is
possible to detect in the visibility amplitude a periodicity that encodes the
shape of the first photon ring. We are able to infer this shape with an
accuracy and precision that is not significantly affected by the level of
instrument noise, up to a certain, rather sharp threshold. Detecting any
signal becomes very challenging past this threshold, which therefore sets a
clear limit on our measurement ability within this class of models and given
our inference technique. Averaging over multiple snapshots can increase this
threshold and restore our ability to detect a photon ring signature even in
the presence of elevated noise or stronger fluctuations.

When analyzing the visibility amplitude in a baseline window where neither the
first nor the second photon ring dominate the signal, it becomes more
challenging to infer a periodicity and hence an interferometric diameter. In
such transition regimes, the periodic ringing of the visibility sometimes
leads one to infer a ring shape that only differs from the “correct” shape
(the one inferred in the absence of noise and fluctuations) by an overall
scaling factor. Fortuitously, this means that one can still infer the correct
fractional asymmetry of the ring, even when its inferred diameter is off by
some (small) factor.

The remainder of this paper is organized as follows. In Sec. II, we briefly
present the phenomenological model we use to simulate time-averaged and time-
dependent data, and the inference scheme we use to extract ring diameters from
a visibility. Then, in Sec. III, we study the impact of instrument noise on
time-averaged images. In Sec. IV, we move on to time-dependent models and
study the effects of astrophysical fluctuations in isolation (in the absence
of instrument noise). Next, in Sec. V, we study a highly challenging case in
the presence of both instrument noise and astrophysical fluctuations, and
demonstrate that a shape measurement still works under adverse conditions.
Finally, we conclude with a summary of our results in Sec. VI, relegating some
details of threshold selection and model characterization to two appendices.

##  II Phenomenological source model

Throughout this paper, we use natural units with $G_{\rm N}$ and $c$ set to
unity. We simulate black hole images using the method of Ref. Gralla  _et al._
(2020), as implemented in AART Cárdenas-Avendaño  _et al._ (2023). In this
section, we briefly review the key ingredients of this model, referring the
reader to Ref. Cárdenas-Avendaño  _et al._ (2023) for further details.

We consider a disk of emitters on circular-equatorial Keplerian orbits. The
disk terminates at an inner edge that corresponds to the innermost stable
circular orbit (ISCO) radius. Past it, the emitters plunge into the hole
following Cunningham’s prescription Cunningham (1975). The intensity at
Cartesian image-plane position $(\alpha,\beta)$ is computed by (analytically)
tracing the corresponding light ray back into the emitting region. The
observed intensity, $I_{\rm o}(\alpha,\beta)$, increases each time the ray
intersects the accretion disk, by an amount determined by the source
emissivity, $I_{\rm s}$. The total observed intensity is then given by
Cárdenas-Avendaño  _et al._ (2023)

$\displaystyle I_{\rm
o}(\alpha,\beta)=\sum_{n=0}^{N-1}\zeta_{n}g^{3}\mathopen{}\mathclose{{}\left(r_{\rm
s}^{(n)},\alpha,\beta}\right)I_{\rm s}\mathopen{}\mathclose{{}\left(r_{\rm
s}^{(n)},\phi_{\rm s}^{(n)},t_{\rm s}^{(n)}}\right),$ (1)

where $x_{\rm s}^{(n)}=x_{\rm s}^{(n)}(\alpha,\beta)$ denotes the
(analytically known) equatorial position where the ray intersects the
equatorial plane for the $(n+1)^{\text{th}}$ time on its backward trajectory
from image-plane position $(\alpha,\beta)$, up to a total number
$N(\alpha,\beta)$ along its maximal extension. The redshift factor $g$ is
determined by the motion of the emitters, and $\zeta_{n}$ is a “fudge” factor,
assumed to be equal to $1$ for $n=0$ and $3/2$ for $n\geq 1$. This factor is
included to account for the effects of the disk’s geometrical thickness Chael
_et al._ (2021); Vincent  _et al._ (2022).

Under this prescription, the observed intensity receives two contributions:
one purely geometric (encoded in the redshift factor $g$), and the other
purely astrophysical (set by the source emissivity $I_{\rm s}$ and fudge
factor $\zeta_{n}$). It is now easy to prescribe time-averaged equatorial
models, as in Refs. Gralla  _et al._ (2020); Paugnat  _et al._ (2022), or
time-dependent ones, as in Ref. Cárdenas-Avendaño  _et al._ (2023).

###  II.1 Time-averaged images

To directly obtain time-averaged images, we assume that the source emissivity
is a function of radius only: $I_{\rm s}(r_{\rm s},\phi_{\rm s},t_{\rm
s})=J(r_{\rm s})$. A convenient parametric form for the radial emission
profile $J(r_{\rm s})$ may be derived from the Johnson’s Standard Unbounded
(SU) distribution,

$\displaystyle J_{\rm
SU}(r;\mu,\vartheta,\gamma)\equiv\frac{e^{-\frac{1}{2}\mathopen{}\mathclose{{}\left[\gamma+\operatorname{arcsinh}\mathopen{}\mathclose{{}\left(\frac{r-\mu}{\vartheta}}\right)}\right]^{2}}}{\sqrt{\mathopen{}\mathclose{{}\left(r-\mu}\right)^{2}+\vartheta^{2}}},$
(2)

in which the three parameters $\mu$, $\vartheta$, and $\gamma$ respectively
control the location of the profile’s peak, its width, and its asymmetry. An
example of a time-averaged image obtained in this fashion is shown in the
upper-left panel of Fig. 1. Throughout this paper, we choose the location of
the peak to be proportional to either the outer or inner event horizon radius,
$r_{\pm}=M\pm\sqrt{M^{2}-a^{2}}$ (see Table 1).

Figure 1: Black hole images (left) and their respective visibility amplitudes
(right) for spin-aligned (blue) and spin-perpendicular (red) cuts across each
image. The black hole spin is $a/M=94\%$ and the observer inclination is
$\theta_{\rm o}=20^{\circ}$. From top to bottom, the magnitude $\sigma$ of the
astrophysical fluctuations increases. In the absence of astrophysical
fluctuations ($\sigma=0$), the image can be understood as being already time-
averaged, with an underlying radial emission profile given by Eq. (2) with the
parameters of $J_{\rm SU}(r_{\rm s})$ corresponding to the profile P1 defined
in Table 1. These visibility amplitudes clearly show that as the amplitude
$\sigma$ of the astrophysical fluctuations grows, it becomes strictly harder
to identify a clear periodicity in the interferometric signal.

###  II.2 Time-dependent models

In physical simulations, source fluctuations arise from various phenomena,
including fluid turbulence, magnetic reconnection, or inherently random
interactions between particles. Here, we model all such fluctuations using the
AART implementation Cárdenas-Avendaño  _et al._ (2023) of the inoisy code Lee
and Gammie (2021), a stochastic model capable of replicating many features of
realistic accretion flows onto a black hole. This code outputs a
spatiotemporally varying emissivity $I_{\rm s}(\mathbf{x}_{\rm s})$ by
generating realizations of a GRF $\hat{\mathcal{F}}(\mathbf{x}_{\rm
s};\mathbf{u}_{i};\lambda_{i})$, where the index
$i\in\mathopen{}\mathclose{{}\left\\{0,1,2}\right\\}$ ranges over multiple
parameters: the vector $\mathbf{u}_{0}$ controls the temporal correlation of
the flow, with characteristic correlation time $\lambda_{0}$, while the
spatial vectors $\mathbf{u}_{1}$ and $\mathbf{u}_{2}$ determine its spatial
structure, with characteristic correlation lengths $\lambda_{1}$ and
$\lambda_{2}$. The GRF is a zero-mean Matérn field obtained by solving a
stochastic partial differential equation Lee and Gammie (2021), and it defines
a source emissivity via Cárdenas-Avendaño  _et al._ (2023)

$\displaystyle I_{\rm s}(\mathbf{x}_{\rm s})=\mathcal{J}(\mathbf{x}_{\rm
s})\equiv J_{\rm SU}(r_{\rm s})e^{\sigma\hat{\mathcal{F}}(\mathbf{x}_{\rm
s})-\frac{1}{2}\sigma^{2}},$ (3)

where $\sigma$ is a parameter that controls the amplitude of the fluctuations.
In this way, the radial emission profile $J_{\rm SU}(r_{\rm s})$ becomes an
envelope for the resulting stochastic source $\mathcal{J}(\mathbf{x}_{\rm
s})$, and one recovers the time-averaged image when $\sigma=0$. Examples of
time-dependent snapshots are displayed in the bottom two rows of Fig. 1. In
all of our simulations, we set $\lambda_{0}=2\pi/\Omega_{\rm K}$, where
$\Omega_{\rm K}$ is the Keplerian orbital frequency, $\lambda_{1}=5r_{\rm s}$,
and $\lambda_{2}=0.1\lambda_{1}$. The three-dimensional unit vectors
$\mathbf{u}_{i}$ are chosen as in Refs. Lee and Gammie (2021); Cárdenas-
Avendaño  _et al._ (2023)—see, e.g., Eqs. (109)-(111) in Ref. Cárdenas-
Avendaño  _et al._ (2023). In particular, we have chosen the major axis
$\mathbf{u}_{1}(\mathbf{x}_{\rm s})$ of the spatial correlation tensor to lie
at a constant angle $\theta_{\angle}=20^{\circ}$ relative to the equatorial
circles of constant radius $r_{\rm s}$, as this produces a spiral arm
structure that is broadly consistent with GRMHD simulations Guan  _et al._
(2009); Lee and Gammie (2021).

###  II.3 Complex visibility and visibility amplitude

Given an image produced using Eq. (1), we compute its radio visibility as the
(complex) Fourier transform

$\displaystyle V(\mathbf{u})=\int I_{\rm o}(\mathbf{x}_{\rm o})e^{-2\pi
i\mathbf{u}\cdot\mathbf{x}_{\rm o}}\mathop{}\\!\mathrm{d}^{2}\mathbf{x}_{\rm
o},$ (4)

where $\mathbf{x}_{\rm o}=(\alpha,\beta)/r_{\rm o}$ are dimensionless
coordinates (or angles) on the sky of a distant observer at large radius
$r_{\rm o}\gg M$, while the sampled baseline $\mathbf{u}$ is a dimensionless
distance between telescopes in the array, projected onto the plane
perpendicular to the line of sight and measured in units of the observational
wavelength. As in Refs. Gralla  _et al._ (2020); Cárdenas-Avendaño  _et al._
(2023), instead of taking the above $2$D Fourier transform, we apply the
projection-slice theorem to compute $|V(u,\varphi)|$, the visibility amplitude
along slices of fixed polar angle $\varphi$ in the Fourier plane. To model
observations of M87∗, we convert the Cartesian coordinates $(\alpha,\beta)$
into angles $\mathbf{x}_{\rm o}$ by using the mass-to-distance ratio
$(M/r_{\rm o})_{\rm M87^{*}}=3.62\,\mu$as.

This effectively fixes the horizontal scale of the snapshot visibilities shown
in Fig. 1. To fix their vertical scale, we normalize the visibilities such
that $|V(0)|=0.6\,$Jy, which corresponds to a total compact source flux of
$0.6\,$Jy Lu  _et al._ (2023).

In the “universal regime” $1/d\ll u\ll 1/w$ of baselines long enough to
resolve a ring of diameter $d$, but not its width $w$, the visibility
amplitude takes the universal form

$\displaystyle|V(u,\varphi)|$
$\displaystyle=\frac{\sqrt{\mathopen{}\mathclose{{}\left(\alpha_{\varphi}^{\rm
L}}\right)^{2}+\mathopen{}\mathclose{{}\left(\alpha_{\varphi}^{\rm
R}}\right)^{2}+2\alpha_{\varphi}^{\rm L}\alpha_{\varphi}^{\rm
R}\sin\mathopen{}\mathclose{{}\left(2\pi d_{\varphi}u}\right)}}{\sqrt{u}},$
(5)

where $d_{\varphi}$ denotes the projected diameter of the ring, while the
functions $\alpha_{\varphi}^{\rm L/R}$ encode its intensity profile Gralla
(2020).

Given a simulated visibility amplitude, we attempt111As we will later see, it
is not always possible to achieve this due to either added instrument noise or
large astrophysical fluctuations. to extract a projected ring diameter
$d_{\varphi}$ at every baseline angle $\varphi$ by fitting $|V(u,\varphi)|$ to
the functional form (5). Then, we can check whether the resulting
$d_{\varphi}$ follows the GR-predicted functional form for a photon ring
Gralla and Lupsasca (2020b); Paugnat  _et al._ (2022),

$\displaystyle\frac{d_{\varphi}}{2}$
$\displaystyle=R_{0}+\sqrt{R_{1}^{2}\sin^{2}\mathopen{}\mathclose{{}\left(\varphi-\varphi_{0}}\right)+R_{2}^{2}\cos^{2}\mathopen{}\mathclose{{}\left(\varphi-\varphi_{0}}\right)}.$
(6)

This expression represents a “circlipse”: a sum of a circle (with radius
$R_{0}$) and an ellipse (with axes $R_{1}$ and $R_{2}$). The phase
$\varphi_{0}$ adds some flexibility to accommodate for rotations of the image.
To extract the diameters $d_{\varphi}$ from Eq. (5), we use the fitting method
described in the next section, which is a version of the procedure presented
in Ref. Paugnat  _et al._ (2022), but generalized to take into account the
effects of instrument noise and astrophysical fluctuations.

###  II.4 Fitting method

Given any profile, we consider $36$ visibility amplitudes computed along
baseline angles
$\varphi\in\mathopen{}\mathclose{{}\left\\{0^{\circ},5^{\circ},\ldots,175^{\circ}}\right\\}$,
where the angle $\varphi\in[0,2\pi)$ also parameterizes $d_{\varphi}$ and
$\alpha_{\varphi}^{\rm L/R}$ Gralla and Lupsasca (2020b). For each baseline
angle $\varphi$, we use SciPy’s simple fitting routine `curve_fit` to fit the
sampled points of the visibility amplitude $|V(u,\varphi)|$ to the functional
form (5), resulting in some best-fit parameters $d_{\varphi}$ and
$\alpha_{\varphi}^{\rm L/R}$.

For this fitting method to work in practice, we must provide suitable bounds
and initial guesses for the three fitting parameters. To set approximate
bounds for $\alpha_{\varphi}^{\rm L/R}$, we use Eq. (5) to derive relations
between their extrema and those of the visibility amplitude $|V(u,\varphi)|$
on the cut:

$\displaystyle\alpha_{\rm max}^{\rm L*}+\alpha_{\rm max}^{\rm R*}$
$\displaystyle\approx V_{\rm max}\sqrt{u_{\rm max}},$ (7)
$\displaystyle\alpha_{\rm max}^{\rm L*}-\alpha_{\rm max}^{\rm R*}$
$\displaystyle\approx V_{\rm min}\sqrt{u_{\rm min}}.$ (8)

Solving these equations yields the approximate bounds

$\displaystyle\alpha_{\rm max}^{\rm L/R}$
$\displaystyle=\frac{1}{2}\mathopen{}\mathclose{{}\left(V_{\rm
max}\sqrt{u_{\rm max}}\pm V_{\rm min}\sqrt{u_{\rm min}}}\right).$ (9)

These vary across the different baseline angles $\varphi$, since the
visibility profile along each cut is generally different. As for the lower
bounds on $\alpha_{\varphi}^{\rm L/R}$, we set them to zero.

For the diameter, we impose the prior $d_{\varphi}\in[35,45]\,\mu$as, since
our analysis is based on M87∗, and photon ring diameters outside this range
are excluded on physical grounds. The initial guess for $d_{\varphi}$ is
always set to $40\,\mu$as and the initial guess for $\alpha_{\varphi}^{\rm
L/R}$ to $k\cdot\alpha_{\rm max}^{\rm L/R}$, where $k$ is some positive
constant less than unity. To fix a value for $k$, we tried several fits with
$k\in\mathopen{}\mathclose{{}\left\\{0.05,0.10,\ldots,0.95}\right\\}$ and
recorded the average root-mean-square deviation (RMSD) for each trial, as
defined in Eq. (10) below. We found that a value of $k=0.15$ consistently gave
good fits, that is, it consistently led to the lowest RMSD.

Having obtained best-fit parameters $d_{\varphi}$ and $\alpha_{\varphi}^{\rm
L/R}$ at every angle $\varphi$, the subsequent parts of the fitting method
closely follow the procedure described in Ref. Paugnat  _et al._ (2022). That
is, we proceed by surveying the multi-peaked distribution of the RMSD, defined
by an average $\mathopen{}\mathclose{{}\left\langle}\right\rangle_{u}$ over
baselines as

$\displaystyle\text{RMSD}_{u}(d)=\frac{\sqrt{\mathopen{}\mathclose{{}\left\langle\mathopen{}\mathclose{{}\left[|V_{\rm
fit}(u;d)|-|V(u)|}\right]^{2}}\right\rangle_{u}}}{\mathopen{}\mathclose{{}\left\langle|V_{\rm
fit}(u;d)|}\right\rangle_{u}}.$ (10)

However, instead of considering an adaptive range around $d_{\varphi}$ with a
width determined by the power associated with the baseline window, as in Ref.
Paugnat  _et al._ (2022), we instead fix the range of candidate ring diameters
to $[8M,12M]$. Again, for M87* profiles, this is justifiable on physical
grounds.

Back in line with Ref. Paugnat  _et al._ (2022), we then survey diameters
$d_{\varphi}$ that maximize the goodness-of-fit $g(d_{\varphi})$, defined as

$\displaystyle g(d_{\varphi})\equiv e^{-\text{RMSD}_{u}(d_{\varphi})},$ (11)

and retain those for which $g(d_{\varphi})$ exceeds some threshold, which,
throughout our analysis, we set to be $g(d_{\varphi})\geq 0.1$.

These candidate angle-dependent diameters $d_{\varphi}$ form multiple distinct
circlipses, which we denote by $\mathcal{C}_{i}$. The $\mathcal{C}_{i}$ are
separated by a gap $\Delta\mathcal{C}_{i}\approx 1/u_{w}$, where $u_{w}$ is
the length of the baseline in the middle of the chosen baseline window. Each
$\mathcal{C}_{i}$ has a joint goodness-of-fit given by

$\displaystyle g(\mathcal{C}_{i})$
$\displaystyle=\prod_{\varphi=0^{\circ}}^{175^{\circ}}g(d_{\varphi}).$ (12)

The most probable circlipse is the one with the largest joint goodness-of-fit
$g(\mathcal{C}_{i})$. We fit it to the GR-predicted functional form (6) for
the photon ring diameter to find the parameters
$\mathbf{R}=\mathopen{}\mathclose{{}\left\\{R_{0},R_{1},R_{2},\varphi_{0}}\right\\}$
that minimize

$\displaystyle\text{RMSD}_{\varphi}(\mathbf{R})=\frac{\sqrt{\mathopen{}\mathclose{{}\left\langle\mathopen{}\mathclose{{}\left[d_{\rm
fit}(\varphi;\mathbf{R})-d_{\varphi}}\right]^{2}}\right\rangle_{\varphi}}}{\mathopen{}\mathclose{{}\left\langle
d_{\rm fit}(\varphi;\mathbf{R})}\right\rangle_{\varphi}}.$ (13)

For various reasons, one must select a narrow baseline window when performing
a fit Paugnat  _et al._ (2022), and the characteristic baseline length $u_{w}$
of the window is especially important. Indeed, $u_{w}$ determines the specific
photon ring whose geometry the fits are probing, since different subrings
dominate different regions of the visibility domain Johnson  _et al._ (2020).

In particular, a choice of baseline window that falls within a transition
region between regimes dominated by one photon subring and the next is less
favorable. In such regions, interference between the signals of the two rings
prevents the emergence of a “clear” interferometric signature Cárdenas-
Avendaño and Lupsasca (2023), thereby inhibiting our ability to recover
information about the shape of either photon ring.

Since we are particularly interested in measurements of the $n=1$ photon ring,
we consider baseline windows in the relatively shorter range between
$u\in[0,150]\,\text{G}\lambda$.222For example, for observations at $345$ GHz,
a baseline of $150\,\text{G}\lambda$ translates to a telescope separation of
about 10 Earth diameters.

This concludes our presentation of the elements needed to investigate
measurements of the first photon ring in the presence of astrophysical
fluctuations and instrument noise. Before proceeding with our analysis, we
briefly summarize the steps in our methodology. First, we ray trace simulated
black hole images using Eq. (1), in either a time-averaged model with no
astrophysical fluctuations or a time-dependent one. For a given image, we
compute its visibility amplitude at various baseline angles $\varphi$ via Eq.
(4). Then, we use the fitting method outlined in Sec. II.4 to extract a ring
shape $d_{\varphi}$ from the visibility amplitude, which we compare to the GR
prediction (6).

In the next sections, we study each effect separately, starting with the
impact of instrument noise, followed by that of fluctuations, and finally, an
example with both.

##  III The impact of instrument noise

In this section, we investigate the effect of instrument noise on the inferred
shape of the first photon ring. In particular, we consider how instrument
noise affects the accuracy and precision of our fitting procedure, and how
much noise our method can handle before breaking down. After finding its
limits, we close this section by discussing which choice of baseline window
might be most conducive to accurate measurements of the first photon ring.

Given a simulated complex visibility (4), we model the effects of instrument
noise by adding to it a realization of a complex Gaussian distribution
$\mathcal{N}_{\mathcal{C}}(0,s)$ with mean zero and standard deviation $s$.

After we add instrument noise, we would like to know how much of an effect it
has on the underlying signal. To understand this, it is instructive to keep
track of the power associated with a given visibility amplitude. For any given
$|V(u)|$, we define the power of the visibility amplitude across a baseline
window $[u_{1},u_{2}]$ as

$\displaystyle
P_{\varphi}(u_{1},u_{2})=\frac{\mathopen{}\mathclose{{}\left|V(u_{1},\varphi)}\right|+\mathopen{}\mathclose{{}\left|V(u_{2},\varphi)}\right|}{2}.$
(14)

This power is a measure of the strength of the visibility amplitude in the
baseline range $[u_{1},u_{2}]$ at fixed angle $\varphi$. Of course, it varies
with the overall normalization of the visibility amplitude. This notion of
power is especially important when thinking about the significance of some
level of noise, since it provides—in combination with $s$—a rough estimate of
the signal-to-noise ratio (SNR).

###  III.1 Accuracy and precision of the inferred interferometric diameters

We begin by assessing the effect of instrument noise on the accuracy and
precision with which we can infer an interferometric ring diameter using our
fitting method.

We consider a Kerr black hole with spin $a/M=94\%$, observed from an
inclination of $\theta_{\rm o}=20^{\circ}$. We choose the time-averaged
emission profile to be given by Eq. (2) with parameters $\mu=r_{-}$,
$\gamma=-3M/2$, and $\vartheta=M/2$. This profile is listed as P1 in Table 1.

Parameter | Value
---|---
$a/M$ | $94\%$
$\theta_{\rm o}$ | $20^{\circ}$
P$1=(\mu,\vartheta/M,\gamma)$ | $\mathopen{}\mathclose{{}\left(r_{-},\frac{1}{2},-\frac{3}{2}}\right)$
P$2=(\mu,\vartheta/M,\gamma)$ | $\mathopen{}\mathclose{{}\left(\frac{3}{2}\,r_{+},1,0}\right)$
Table 1: The black hole spin and inclination of the example case we consider
throughout this paper, and the parameters for our two radial emission profiles
(2). The outer/inner event horizon radii are denoted by
$r_{\pm}=M\pm\sqrt{M^{2}-a^{2}}$.

We carry out $500$ fits with distinct noise realizations, first with a noise
level of $s=1.5\,$mJy and then again with $s=2.5\,$mJy, across the baseline
window $[86,116]\,\text{G}\lambda$.333For a noise level of $s=1.5\,$mJy, Fig.
7 shows that this baseline window gives the lowest average RMSD over all the
profiles we study in this work (see Sec. III.3 for details). Furthermore, as
we will see shortly, inferring an interferometric diameter across this
particular baseline window is quite challenging when using the astrophysical
parameters of the profile P1 listed in Table 1.

Over the $36$ baseline angles $\varphi$ that we examine within this window,
the visibility amplitude attains a minimum power (14) of $0.56\,$mJy, a
maximum value of $1.18\,$mJy, and an average value of $0.84\,$mJy. As shown in
the right panels of Fig. 4, the instrument noise levels $s=1.5\,$mJy and
$s=2.5\,$mJy really do affect the visibility amplitude and its underlying
interferometric signal.

For each successful fit, we compute the maximal (and orthogonal) ring
diameters

$\displaystyle d_{\parallel}$ $\displaystyle=2(R_{0}+R_{1}),$ (15)
$\displaystyle d_{\perp}$ $\displaystyle=2(R_{0}+R_{2}),$ (16)

where $R_{0}$ and $R_{2}$ are the best-fit parameters in Eq. (6). When
$\varphi_{0}=0$, these are exactly the diameters parallel and perpendicular to
the projected black hole spin axis.

Together, they define the fractional ring asymmetry

$\displaystyle f_{A}=1-\frac{d_{\perp}}{d_{\parallel}},$ (17)

which characterizes the “shape” of the circlipse.

Figure 2: The multi-peaked distribution of inferred photon ring diameters in
the absence of noise or astrophysical fluctuations, for a black hole with spin
$a/M=94\%$ surrounded by the emission profile P1 specified in Table 1 and
observed from an inclination of $\theta_{\rm o}=20^{\circ}$. For each baseline
angle $\varphi$, we plot the possible inferred diameters with dots whose sizes
are set by the relative magnitudes of their goodness-of-fit (and not their
absolute values). That is, the point sizes should only be compared vertically
(i.e., at fixed $\varphi$) in order to identify which diameter had the highest
goodness-of-fit, the second highest, and so on, at that particular baseline
angle. The dashed black line is the circlipse that best fits the critical
curve, while the background shaded grey lines are the circlipses inferred from
the same black hole, but with the $30$ different astrophysical profiles
specified by the parameters in Table 2. This figure and the upper-left panel
of Fig. 4 both display the same data, including the same inferred circlipse
(blue curve). However, here we use point sizes rather than colors to better
illustrate the differences in the goodness-of-fits of the inferred diameters
and to highlight the “jumps” in the best-fit diameters that occur at
$d_{\varphi}\approx 9.2M$ and $d_{\varphi}\approx 9.8M$. Figure 3: The
visibility amplitude corresponding to the direct image ($n=0$, green-dotted
lines), the first two photon rings ($n=1$, orange-dashed lines, and $n=2$,
blue-dash-dotted lines), and the total image ($0\leq n\leq 2$, black solid
lines) for the profile P1 (see Table 1), in the absence of instrument noise
and astrophysical fluctuations. The pink-shaded region corresponds to the
baseline window $[86,116]\,\text{G}\lambda$ over which we attempt to infer an
interferometric diameter. The left and right panels display the visibility
amplitude at baseline angles $\varphi=0^{\circ}$ and $\varphi=125^{\circ}$,
respectively. The right panel inset shows the corresponding image. In this
baseline window, the visibility undergoes at every angle $\varphi$ a
transition between regimes dominated by the $n=1$ and $n=2$ photon rings.
Nevertheless, one can still infer a circlipse shape for $d_{\varphi}$, as
shown in Figs. 2 and 4 (upper-left panel).

As described in Sec. II.4, for each fitting attempt, we generate multiple
circlipses $\mathcal{C}_{i}$ by surveying the multi-peaked distribution of the
RMSD (10), and then we fit the circlipse with the highest joint goodness-of-
fit (12) to the expected functional form (6) for the photon ring shape in
order to infer its angle-dependent diameter $d_{\varphi}$.

We draw the best-fit circlipse in the absence of noise and fluctuations as a
blue line in Fig. 2 and the upper-left panel of Fig. 4. The first shows the
relative distribution of the goodness-of-fits for the inferred diameters,
while the second offers a more faithful representation across angles. This
facilitates a comparison of the joint goodness-of-fits across all the
candidate circlipses.

We see in Fig. 2 that for the emission profile P1 given in Table 1, which is
broadly consistent with the EHT results of M87∗ on Earth baselines Gralla  _et
al._ (2020), the inferred diameters $d_{\varphi}$ with the highest goodness-
of-fit can “jump” between two circlipses at certain angles $\varphi$. These
discontinuities arise as a result of the choice of baseline window, and their
presence and location depend on the set of astrophysical parameters under
consideration. In the chosen window, the visibility amplitude of profile P1 is
_always_ sampled (at every angle $\varphi$) in the transition region between
the regimes dominated by the first and second photon rings (see Fig. 3). That
is, for all of the $36$ sampled baseline angles, the first photon ring never
dominates the signal across the entire baseline window in which we perform the
fits. As discussed in detail in Ref. Paugnat  _et al._ (2022), a diameter
inferred in such a setting lacks a very sharp geometric interpretation: if
neither the $n=1$ nor the $n=2$ photon ring dominates the visibility
amplitude, then its ringing is not directly related to the physical size of
either ring; rather, it is a feature of the continuity of the signal and its
periodicity Paugnat  _et al._ (2022). In other words, when sampling in a
transition region, one can still infer an interferometric diameter
$d_{\varphi}$ by fitting the universal form (5), but this diameter lacks a
clear interpretation in the image.

Despite this jump in the inferred diameter $d_{\varphi}$, one can still (in
some cases) infer a “correct” circlipse shape, as shown for instance in the
lower panel of Fig. 4 of Ref. Cárdenas-Avendaño and Lupsasca (2023). (The
panel shows that a “good” circlipse can be inferred in the baseline window
$[70,100]\,\text{G}\lambda$, but not the window $[40,70]\,\text{G}\lambda$,
for the emission profile P2 given in Table 1). This is not too surprising
because the visibility amplitude is dominated by a ring on either side of the
transition region, so its ringing encodes a circlipse shape on both sides of
the transition: by continuity, we are then led to expect the interferometric
diameter within the transition region to also follow a circlipse shape.

Indeed, the circlipse shape we infer in this case is still the “correct” one,
that is, the one closest to the circlipse shape associated with the critical
curve (the dashed black line in Fig. 2). As we vary the underlying
astrophysical profile $J_{\rm SU}(r_{\rm s})$ across the set of $30$
parameters specified in Table 2, we continue to infer similar circlipse shapes
that cluster in a band around the same location (see the shaded grey
background lines in Fig. 2).

Figure 4: The multi-peaked distribution of inferred photon ring diameters for
a black hole with spin $a/M=94\%$ surrounded by the emission profile P1
specified in Table 1 and observed from an inclination of $\theta_{\rm
o}=20^{\circ}$. The four panels display simulated measurements with various
combinations of instrument noise and astrophysical fluctuations across the
window $[86,116]\,\text{G}\lambda$: no noise or fluctuations (top left),
$s=1.5\,$mJy of instrument noise alone (top right), astrophysical fluctuations
with $\sigma=0.25$ alone (bottom left), and a combination of both instrument
noise and astrophysical fluctuations with $s=1.5\,$mJy and $\sigma=0.25$,
respectively (bottom right). In the absence of astrophysical fluctuations
($\sigma=0$), realizations of the complex Gaussian noise
$\mathcal{N}_{\mathcal{C}}(0,s)$ are added to the complex visibility across
each baseline angle $\varphi$. In the presence of fluctuations ($\sigma>0$),
we carry out the same process separately across $N=10$ snapshots, before
taking their coherent time average. In either case, we then fit the sampled
visibility amplitude across all angles to the GR-predicted functional form (5)
to obtain the ring diameter $d_{\varphi}$ that locally minimizes the RMSD
(10). We also consider as possible diameters the other local minima of the
RMSD (or, equivalently, the local maxima of the goodness-of-fit) in the range
$[8M,12M]$. Each panel has its own color scale, whose limits are set by the
absolute maximum and minimum of the goodness-of-fit for all the inferred
diameters across all baseline angles. The multi-peaked diameter distribution
gives rise to multiple circlipse shapes, and the blue lines correspond to the
circlipse with the highest joint goodness-of-fit (12) in each fitting attempt.

As shown in the upper-right panel of Fig. 4, we are still able to infer a
circlipse in the correct band, even using noisy observations in a transition
region. This circlipse, however, is slightly distorted and displaced relative
to the one inferred in the absence of noise. Since each of the $500$ fitting
attempts in this survey has a different realization of noise, we expect the
circlipses inferred from each fit to be deformed in slightly different ways.
Indeed, the different diameters $d_{\varphi}$ inferred from each fit produce
the distribution shown in Fig. 5. For example, the inferred parallel diameter
$d_{\parallel}$ is given, for each realization of noise, by the ordinates of
the blue curves at $\varphi=90^{\circ}$ in Fig. 4.

For each realization of noise, the inferred circlipse is slightly displaced
vertically, leading to small differences in the inferred values of
$d_{\parallel}$ across the $500$ fits.

Figure 5: The inferred-diameter distributions for the parallel diameter
$d_{\parallel}$ (left) and the perpendicular diameter $d_{\perp}$ (right), for
a black hole with spin $a/M=94\%$ surrounded by the emission profile P1
specified in Table 1 and observed from an inclination of $\theta_{\rm
o}=20^{\circ}$. The fits are performed in the baseline window
$[86,116]\,\text{G}\lambda$ with $500$ realizations of instrument noise for
$s=1.5\,$mJy (green histograms) and another $500$ for $s=2.5\,$mJy (purple
histograms). The presence of instrument noise shifts the centers of the
distributions relative to the diameters inferred in the absence of noise
(i.e., $s=0\,$mJy), denoted by the vertical dashed lines. For the inferred
parallel diameters, the centers of the $s=1.5\,$mJy and $s=2.5\,$mJy
histograms are shifted by $0.25\%$ and $0.46\%$, respectively, relative to the
inferred $d_{\parallel}$ in the absence of noise. Similar shifts, but in the
opposite direction, occur for the inferred $d_{\perp}$ histograms. As
expected, as the level of noise increases, the widths of the histograms also
increase. In particular, as the noise level grows from $s=1.5\,$mJy to
$s=2.5\,$mJy, the widths of the $d_{\parallel}$ and $d_{\perp}$ histograms
increase by factors of $\approx 1.3$, and $\approx 1.1$, respectively. The
most pronounced effect of increased instrument noise is a higher rate of fit
failure (i.e., a decrease in the area of the histograms): $95\%$ of the fits
are successful when $s=1.5\,$mJy, while only $25\%$ of fits succeed when
$s=2.5\,$mJy.

The distribution of diameters shown in Fig. 5 is to be distinguished from the
multi-peaked distribution of diameters presented in Fig. 4; the latter is due
to the RMSD having various local minima, each of which may or may not
correspond to the global minimum, while the former is due to different
realizations of instrument noise, each of which leads to an inferred circlipse
that is the same as in the noiseless ($s=0\,$mJy) case, but slightly displaced
and distorted by the effects of noise.

To assess the accuracy of these diameter distributions, we use as a benchmark
the diameters $d_{\parallel}$ and $d_{\perp}$ inferred in the noiseless case
($s=0\,$mJy), which are plotted as dashed vertical lines in Fig. 5. Although
these diameters do not correspond to the interferometric diameters of the
$n=1$ photon ring (since the visibility amplitude is in a region of transition
at all baseline angles), nonetheless they still offer insight into the
accuracy of our diameter inference. This is because, as Fig. 2 shows, the
circlipse they correspond to is remarkably close to the shape of the critical
curve. In particular, the diameters $d_{\parallel}$ and $d_{\perp}$ inferred
in the absence of noise differ from those of the critical curve by $0.58\%$
and $0.61\%$, respectively.

Moreover, the RMSD of the fit to the functional form of the circlipse is
$0.008\%$ in the absence of noise. The close proximity of the inferred
circlipse to the critical curve shape, together with the low RMSD of the fit,
suggest that, at least in the absence of noise and astrophysical fluctuations,
our fitting scheme is capable of recovering information about the underlying
geometry, even from a transition region. As such, we can use these diameters
as a benchmark to assess the accuracy of the distributions of diameters
inferred in the presence of instrumental noise (and astrophysical
fluctuations).

Figure 6: Average RMSD values (left) and inferred diameters (right) for $15$
fitting attempts across the baseline window $[86,116]\,\text{G}\lambda$ in the
presence of instrument noise ranging from a level $s=10^{-5}\,$Jy to
$s=10^{0}\,$Jy. The vertical lines indicate the noise levels $s=1.5\,$mJy
(green dashed line) and $s=2.5\,$mJy (purple dash-dotted line), and the
horizontal lines on the right panel indicate the diameters inferred in the
absence of noise using the envelope-fitting method of Ref. Paugnat  _et al._
(2022). At the largest plotted noise level of $s=3.2\,$mJy, the average
inferred diameters $d_{\parallel}$ (red crosses) and $d_{\perp}$ (blue dots)
differ by $0.54\%$ and $0.44\%$, respectively, from those inferred in the
absence of noise. For noise levels above $s=3.2\,$mJy, the RMSD plateaus and
successful fits become much less frequent: for $s\leq 3.2\,$mJy, $93\%$ of all
the fitting attempts succeed, whereas only $1\%$ succeed for $s>3.2\,$mJy.

As Fig. 5 shows, adding noise causes the mean of each histogram to shift
slightly relative to the dashed vertical lines showing the diameters inferred
in the absence of instrument noise. In particular, for $d_{\parallel}$, the
means of the $s=1.5\,$mJy and $s=2.5\,$mJy histograms are displaced by
$0.25\%$ and $0.46\%$, respectively, relative to the value of $d_{\parallel}$
inferred in the absence of noise, with shifts of comparable size for
$d_{\perp}$. Moreover, the width of the histograms for $d_{\parallel}$ (or
$d_{\perp}$) increases by a factor of $\approx 1.3$ (or $\approx 1.1$) as the
noise level increases from $s=1.5\,$mJy to $s=2.5\,$mJy.

The changes we observe in the mean values and widths of the histograms in Fig.
5 are quite modest relative to the changes in their areas. In other words, the
strongest effect of increased noise levels is an increased failure rate:
$95\%$ of the fits are successful when this level is $s=1.5\,$mJy, but that
figure decreases to $25\%$ when the noise grows to $s=2.5\,$mJy. As such,
increasing the level of instrument noise leads to a significant reduction in
the likelihood of a successful fit; on the other hand, for successful fits,
increasing the noise level only leads to a modest reduction in the accuracy
and precision of the fit. In sum, the measurements of $d_{\varphi}$ are either
good, or fail.

Since Fig. 5 includes the full range of diameters $d_{\varphi}$ inferred from
our survey, we see that increasing the level of instrument noise never leads
us (in any of the $500$ fits) to select the second circlipse, which exhibits
jumps in $d_{\varphi}$.

###  III.2 Breakdown of our fitting method

Having assessed the accuracy and precision of our fits in the presence of a
few levels of instrument noise, we now vary the noise level $s$ to find out
how much of it our fitting method can handle before breaking down.

We perform fits to the visibility amplitude of a black hole with profile P1
(defined in Table 1) in the baseline window $[86,116]\,\text{G}\lambda$, in
the presence of $15$ realizations of noise and for $250$ different noise
levels chosen to grow exponentially from $s=10^{-5}\,$Jy to $s=1\,$Jy. The
results are shown in Fig. 6, wherein the ordinate of each point in the left
(or right) plot is the average RMSD (or inferred diameter) of the $15$ fits
attempted at the corresponding level of noise. The dashed vertical lines
denote the noise levels $s=1.5\,$mJy and $s=2.5\,$mJy from the last section.

The last points plotted in Fig. 6 correspond to a noise level $s_{\rm
max}=3.2\,$mJy, beyond which fewer than three successful fits can be obtained
for any given $s$. Indeed, the overwhelming majority of fitting attempts above
this level of noise fail:444The very last successful fit we obtained had a
noise of $s=0.53\,$Jy. The diameters $d_{\parallel}$ and $d_{\perp}$ that we
inferred differed by $0.54\%$ and $0.28\%$, respectively, from those inferred
in the absence of noise. for $s\leq s_{\rm max}$, $93\%$ of the fitting
attempts are successful, whereas only $1\%$ are successful for $s>s_{\rm
max}$. The particular level of noise $s=3.2\,$mJy is only significant insofar
as it relates to the power of the visibility amplitude across
$[86,116]\,\text{G}\lambda$: the ratios of $s_{\rm max}$ to the maximum and
average powers are $s_{\rm max}/P_{\rm max}=2.7$ and $s_{\rm max}/P_{\rm
avg}=3.8$. In other words, the success rate of the fits drops below $20\%$
when the noise level $s$ exceeds $3.8$ times the (average) power.

As the left panel of Fig. 6 shows, the RMSD starts to grow significantly
around $s=0.1\,$mJy. By contrast, the inferred diameters remain relatively
insensitive to this depreciation of the quality of fit (as suggested by a
higher RMSD). Indeed, the percentage change in the diameters remains modest
over the entire range of noise levels in Fig. 6. In particular, for
$s=3.2\,$mJy (the highest noise value plotted), the average inferred
$d_{\parallel}$ and $d_{\perp}$ differ from their fitted values in the absence
of noise by only $0.54\%$ and $0.44\%$, respectively.

This small variation of the inferred diameters $d_{\varphi}$ with respect to
the noise level explains the shifts observed in the histogram peaks in Fig. 5.
For the largest values of $s$ in Fig. 6, the inferred diameters asymptote to a
single value. For even larger values of $s$ beyond those shown in Fig. 6, the
inferred shape becomes more circular, which corresponds to the flattening of
the inferred circlipse.

This can be visually understood from the upper-right panel of Fig. 4: when
there is a significant increase in the noise, the inferred diameters start to
oscillate around a common value. Consequently, the ring shape and its
asymmetry get washed out, and one can only infer a straight line corresponding
to a circular radius.

Remarkably, _all_ the diameters inferred in this survey lie within the range
delineated by the two horizontal dashed lines in Fig. 6 (which denote the fits
obtained for $s=0\,$mJy using the envelope fitting method in Ref. Paugnat  _et
al._ (2022)), including the fits obtained sporadically at $s>s_{\rm max}$.
That is, despite the jump in the inferred diameters $d_{\varphi}$ with
$s=0\,$mJy (displayed in Fig. 2), the addition of substantial levels of
instrument noise does not lead us to infer a different circlipse for any of
the noise realizations.

Together, Figs. 5 and 6 suggest that as the noise level increases, the RMSD
also increases, so that the fits fail with higher frequency, as one might
expect. However, the resulting change in the distribution of the inferred
diameters is, in relative terms, modest. Thus, the main effect of increased
instrument noise is a higher failure rate for measurements of the signal
periodicity, but only a modest reduction in their accuracy and precision.

Above a certain noise threshold ($s_{\rm max}=3.2\,$mJy in our survey), the
ability to reliably infer a periodicity from the signal becomes severely
limited (i.e., for $s\geq s_{\rm max}$, the success rate for fitting attempts
drops below $20\%$). Finally, the presence of instrument noise _alone_ does
not exacerbate transition-region effects nor lead one to infer a “wrong”
circlipse.

In a real measurement, even if the noise level exceeds this threshold, it does
not necessarily follow that the shape of the first photon ring is absent from
the signal. It may still be possible to extract the shape information using a
more sophisticated method of data analysis that uses multiple observations and
goes beyond the simple fitting method that is implemented herein.

###  III.3 Suitable baseline windows

To assess which baseline window is most suitable for a photon ring shape
measurement, we now examine several profiles and fit their visibility
amplitudes over various baseline windows, and then analyze the corresponding
distribution of RMSDs.

More specifically, we perform fits using $270$ different time-averaged models,
in which we vary the parameters of the underlying geometry (namely, the black
hole spin and the observer inclination) and the three parameters
$(\mu,\vartheta,\gamma)$ that characterize the radial emission profile (2).
The precise values that we consider for these parameters are presented in
Table 2. We systematically examine a set of baseline windows starting at
$\mathopen{}\mathclose{{}\left\\{0,2,4,\ldots,100}\right\\}\,\text{G}\lambda$
and extending over a range of
$\mathopen{}\mathclose{{}\left\\{5,10,15,\ldots,50}\right\\}\,\text{G}\lambda$.

Parameter | Values
---|---
$a/M$ | $\mathopen{}\mathclose{{}\left\\{0\%,50\%,94\%}\right\\}$
$\theta_{\rm o}$ | $\mathopen{}\mathclose{{}\left\\{10^{\circ},20^{\circ},30^{\circ}}\right\\}$
$\mu$ | $\mathopen{}\mathclose{{}\left\\{r_{-},\frac{1}{2}r_{+},r_{+},\frac{3}{2}r_{+},2r_{+}}\right\\}$
$\vartheta/M$ | $\mathopen{}\mathclose{{}\left\\{\frac{1}{2},1}\right\\}$
$\gamma$ | $\mathopen{}\mathclose{{}\left\\{-1,0,1}\right\\}$
Table 2: The $270$ profiles whose time-averaged images we survey in Fig. 7 in
search for the optimal baseline window for a shape measurement of the first
photon ring. We consider three black hole spins, three observer inclinations,
and $30$ emission profiles. This wide range of parameters accounts for many
possible astrophysical scenarios in which the emission profile peaks at
different locations and exhibits different levels of broadness and steepness.
Although not all of these profiles are realistic, the parameter space is large
enough to encompass a wide range of plausible astrophysical sources. The
outer/inner event horizon radii are denoted by
$r_{\pm}=M\pm\sqrt{M^{2}-a^{2}}$.

This results in $51\times 10=510$ baseline windows. For each one of them, we
perform $4$ fits per profile, each time adding a different realization of
noise with $s=1.5\,$mJy or $s=2.5\,$mJy, producing a total of $1,101,600$
attempted fits. For each profile across a given baseline window, we
conservatively take the maximum RMSD of the $4$ fitting attempts.555When
fitting in a given baseline window, sometimes a periodicity cannot be
recovered from the visibility, so the fit fails and no RMSD is computed. In
some baseline windows (depicted by black cells in Fig. 7), _all_ of the fits
fail. The result of this survey is presented in Fig. 7, where each baseline is
represented by a cell whose color indicates the mean value of the $270$ RMSD
maxima (a fairly conservative estimator).

Figure 7: The distribution of the maximum value of the averaged RMSD obtained
from the fits for the $270$ profiles given in Table 2, in the presence of
instrument noise levels $s=1.5\,$mJy (left) and $s=2.5\,$mJy (right), across
baselines windows $[u_{1},u_{1}+\Delta u]\,\text{G}\lambda$, where
$u_{1}\in\mathopen{}\mathclose{{}\left\\{0,2,\ldots,100}\right\\}$ and $\Delta
u\in\mathopen{}\mathclose{{}\left\\{5,10,\ldots,50}\right\\}$. The dashed
purple cells denote the baseline windows $[86,116]\,\text{G}\lambda$ and
$[88,138]\,\text{G}\lambda$, which have the lowest average worst-fit RMSD
value for the cases $s=1.5\,$mJy and $s=2.5\,$mJy, respectively. The black
cells correspond to baseline windows in which no fit is ever successful. The
white contours correspond to a mean RMSD value of $0.05\%$, and they enclose
$38\%$ and $19\%$ of the baseline windows in the cases $s=1.5\,$mJy and
$s=2.5\,$mJy, respectively.

The cells highlighted with dashed purple borders in Fig. 7 correspond to the
baseline windows $[86,116]\,\text{G}\lambda$ and $[88,138]\,\text{G}\lambda$,
which have the lowest average RMSD when $s=1.5\,$mJy or $s=2.5\,$mJy,
respectively. The average RMSD of the fits across the window
$[86,116]\,\text{G}\lambda$ is $0.012\%$, while the average RMSD across
$[88,138]\,\text{G}\lambda$ is $0.019\%$. The black cells indicate a complete
inability to recover a fit: in the baseline windows corresponding to those
cells, every fitting attempt is unsuccessful. The white solid lines are
contours corresponding to a mean RMSD value of $0.05\%$, and they enclose
$38\%$ and $19\%$ of the baseline windows in the cases $s=1.5\,$mJy and
$s=2.5\,$mJy, respectively.

To demonstrate the qualitative difference in the quality of the fits just
above and below the (arbitrary) RMSD threshold of $0.05\%$, we present in App.
A examples of fits performed within the baseline windows highlighted with
green and red borders in the left panel of Fig. 7.

We can see in Fig. 7 that the more favorable baseline windows (i.e., those
with a lower RMSD) are generally concentrated in the upper-right section of
the heatmaps. In other words, longer baselines and larger window sizes lead to
better shape measurements.

The preference for larger window sizes is explained by the oscillatory
behavior of the visibility amplitudes. Each measurement requires finding the
parameters that best fit an oscillatory function to some noisy data set. Over
a shorter window, and in the presence of noise, a larger proportion of the
parameter space is consistent with the observed signal. By contrast,
observations over longer windows impose stronger constraints on the parameters
of the fit, which must reproduce an increasingly specific oscillatory (and
decaying) pattern over a larger domain.

The preference for longer baselines (within this region of the visibility
domain and for the profiles considered) can be understood in two ways. First,
as the visibility amplitude decreases rapidly on baselines close to $0\,$mJy,
fits on such baselines frequently fail (e.g., for both levels of noise, all
the fits at baseline windows starting at $0\,\text{G}\lambda$ completely
fail). Second, as the baseline length increases, the visibility amplitude
enters the universal photon ring regime and thus becomes increasingly
independent of the astrophysical profile Johnson  _et al._ (2020); Gralla  _et
al._ (2020), resulting in sharper probes of the underlying geometry.

However, when the baseline length increases too much, the steady decay of the
power in the visibility amplitude leads to a decreased SNR, which brings down
the quality of fits. All else being equal, instrument noise generally has a
stronger effect on larger baselines: by Eq. (5), the power decreases roughly
like $\sqrt{u}$ with increasing baseline, so the SNR decreases like
$\sqrt{u}$. Since we only consider baselines extending up to
$150\,\text{G}\lambda$ in this survey, this effect is not substantial, but if
one were to consider even longer baselines, then a baseline-dependent noise
level might be more appropriate than the fixed levels considered here.

Since SNR generally decreases with increasing baseline, it makes sense that
regions of total failure appear in the “better” (longer) baseline windows in
the $s=2.5\,$mJy case before appearing in the “worse” (shorter) windows. As
the noise level $s$ increases, we expect its effect to be strongest on those
baselines where the SNR is smallest, that is, on longer baselines such as
those appearing in the upper-right region of Fig. 7.

The substantial increase in the fitting failure rate as the noise level grows
from $s=1.5\,$mJy to $s=2.5\,$mJy is consistent with Fig. 5, and is a
recurring feature in the surveys described in the rest of this paper. Despite
the significant failure rate at $s=2.5\,$mJy, there are still some baseline
windows in the upper-right region of the plot whose corresponding RMSDs are
comparable to the fits at noise level $s=1.5\,$mJy. Indeed, the maximum RMSD
for the fits across the window $[88,138]\,\text{G}\lambda$ with noise level
$s=2.5\,$mJy is only $0.007\%$ larger than the maximum RMSD for the fits
across the window $[86,116]\,\text{G}\lambda$ with noise level $s=1.5\,$mJy.

This is another recurring theme of our survey: in most baseline windows, as
the instrument noise increases, the likelihood of detecting a periodic signal
decreases, but, provided that it is detectable, its measurement accuracy does
not vary much with the noise level.

As previously indicated, our chosen threshold RMSD of $0.05\%$ indicated by
the white contours in Fig. 7 is somewhat arbitrary (see App. A). We consider
this choice to be rather conservative in the sense that it corresponds to
qualitatively good circlipse fits for the photon ring shape. In a real space
mission, the actual target RMSD value would depend on the science
requirements.

If one can ignore orbital constraints and choose any baseline window to
observe in, then Fig. 7 indicates which windows are better suited to accurate
measurements of the photon ring, _on average_ , for the $270$ profiles that we
consider here. This does not mean, for example, that in the presence of
$s=1.5\,$mJy of instrument noise, all $270$ profiles considered are optimally
observed across the window $[86,116]\,\text{G}\lambda$. Indeed, for some
profiles, a different baseline window might be optimal. For the profile P1
(see Table 1) considered throughout this paper, for example, the visibility
amplitude in the window $[86,116]\,\text{G}\lambda$ lies in a transition
region between regimes dominated by the $n=1$ and $n=2$ rings. As such, a
shorter baseline window may be preferable in order to avoid transition
effects. Besides, Fig. 7 does not indicate the absolute quality of the fits
across each window. If a near-future experiment were to limit us to narrower
baseline windows starting on shorter baselines, then according to Fig. 6,
accurate shape measurements of the photon ring would still be possible: the
least favorable regions of Fig. 7 have RMSDs comparable to those at the upper
range shown in the left panel of Fig. 6, and as the left panel shows, fits
with these RMSDs have inferred diameters that remain close to those inferred
in the absence of noise.

Thus far, we have only studied how instrument noise alone affects the
extraction of a photon ring shape from the visibility amplitude of time-
averaged images. Next, we study the effects of astrophysical fluctuations on
our ability to infer the photon ring shape.

##  IV Impact of astrophysical fluctuations

We begin this section by considering how the scale of astrophysical
fluctuations, which is characterized by the parameter $\sigma$ in Eq. (3), can
affect our ability to extract an interferometric diameter in visibility space.
We then consider the effect of the number $N$ of “snapshots” over which we
time average the visibility. This averaging can be carried out in one of two
ways: either coherently, or incoherently. Coherent averages retain phase
information by first averaging the complex visibility and then taking the
amplitude, whereas incoherent averages discard the relative phases by directly
averaging at the level of the visibility amplitude.

As $N\rightarrow\infty$, astrophysical fluctuations wash out of the average
over snapshot images, and thus one recovers the time-averaged image that
corresponds to the radial emission profile (2) with the same $J_{\rm
SU}(r_{\rm s})$ parameters. A coherently averaged visibility corresponds to
the visibility of this time-averaged image. Experimentally, however, our
current technology limits us to incoherent averaging.

###  IV.1 Impact of the number of snapshots averaged

We consider two emissivity profiles, six magnitudes $\sigma$ of astrophysical
fluctuations, and various spin-inclination combinations specified in Table 3,
comprising a total of $108$ different profiles. For each of these profiles, we
ray trace a simulated black hole movie with a duration of $1000M$, sampled
every $10M$ to produce a total of $N=100$ snapshots. We present the
distribution of the power across the baseline window
$[86,116]\,\text{G}\lambda$ over all considered models in App. B.

Parameter | Values
---|---
$a/M$ | $\mathopen{}\mathclose{{}\left\\{2\%,50\%,94\%}\right\\}$
$\theta_{\rm o}$ | $\mathopen{}\mathclose{{}\left\\{10^{\circ},20^{\circ},30^{\circ}}\right\\}$
$\sigma$ | $\mathopen{}\mathclose{{}\left\\{0.0,0.05,0.15,0.25,0.35,0.45}\right\\}$
P$1=(\mu,\vartheta/M,\gamma)$ | $\mathopen{}\mathclose{{}\left(r_{-},\frac{1}{2},-\frac{3}{2}}\right)$
P$2=(\mu,\vartheta/M,\gamma)$ | $\mathopen{}\mathclose{{}\left(\frac{3}{2}r_{+},1,0}\right)$
Table 3: Parameters for the $108$ movies with astrophysical fluctuations. For
each of the profiles in Table 1, we consider three black hole spins, three
observer inclinations, and six fluctuation scales $\sigma$. When $\sigma=0$,
the source (3) reduces to the time-averaged model (2) with the same parameters
for the radial emission profile $J_{\rm SU}(r_{\rm s})$. The outer/inner event
horizon radii are denoted by $r_{\pm}=M\pm\sqrt{M^{2}-a^{2}}$.

For each of the $108$ profiles, and for $N=5$, $10$, $20$, and $70$ snapshots,
we conduct $10$ fits, each time recording the maximum (i.e., worst-fit) value
of the RMSD. In each case, we randomly choose $N$ snapshots from the total of
$100$, take either their coherent or incoherent average, and then attempt a
fit to the resulting visibility amplitude.

The distributions of the worst-fit RMSD are shown in Fig. 8 for the different
values of $N$, for both coherent (left panel) and incoherent (right panel)
averages. Since these distributions are very similar, it follows that the
phase variations among the images are relatively small, that SNR is high, and
that the strong signal (large amplitude) features are dominating the snapshot
images.

Figure 8: The distributions of the worst-fit RMSD among $10$ fits carried out
for the $108$ time-dependent profiles given in Table 3, when taking the
coherent (left), and incoherent (right) average of (from top to bottom) $N=5$
(purple), $10$ (green), $20$ (blue), and $70$ (red) snapshots. The vertical
dashed line denotes an RMSD of $0.05\%$. As the number of snapshots $N$
increases, the RMSD of the fits generally decreases (i.e., the histograms
shift towards the left) and the number of successful fits increases.

As expected, increasing the number $N$ of snapshots that we average over
causes the RMSD distributions in Fig. 8 to shift towards lower values. In
particular, for the coherent averages, we find that $30\%$ of the profiles
have RMSD values below our chosen threshold of $0.05\%$ (drawn as a vertical
dashed line) when $N=5$. At the other extreme, when $N=70$, this fraction
grows to $59\%$. As expected, we observe a similar trend for the incoherent
averages.

Increasing the number $N$ of averaged snapshots not only lowers the worst-fit
RMSD, but also improves our ability to detect a periodicity: for the coherent
averages, the percentage of successful fits is $53\%$, $55\%$, $61\%$, and
$66\%$ when the snapshots averaged over number $N=5$, $N=10$, $N=20$, and
$N=70$, respectively, with similar values for the incoherent average.

These results are unsurprising and consistent with the fact that in the
$N\to\infty$ limit, one recovers a time-averaged image dominated by a single
persistent feature: the photon ring. Increasing the number $N$ of snapshots
lessens the impact of astrophysical fluctuations, reduces the RMSD of the
fits, and enhances our ability to recover a periodicity. However, the precise
value of $N$ such that the RMSD dips below a certain threshold depends on the
specific details of the fluctuations.

###  IV.2 Impact of the scale of astrophysical fluctuations

Next, we survey the impact that the magnitude $\sigma$ of the astrophysical
fluctuations has on our ability to infer an interferometric diameter. For this
survey, we keep all the other parameters fixed and vary only $\sigma$.

We consider images of a black hole surrounded by the profile P1 given in Table
1, and carry out $500$ fits across the baseline window
$[86,116]\,\text{G}\lambda$, first with $\sigma=0.25$, and then again with
$\sigma=0.45$. Each fit is performed on a visibility amplitude obtained by
averaging over $N=10$ snapshots randomly chosen during each fitting attempt.
We show the distributions of the inferred diameters $d_{\parallel}$ and
$d_{\perp}$ resulting from coherent and incoherent averaging as the solid and
dashed lines in Fig. 9, respectively. In these plots, the vertical dashed
lines are the diameters inferred in the case $\sigma=0$ (i.e., with no
astrophysical fluctuations), which are the same as those in Fig. 5.

Figure 9: The multi-peaked distributions of diameters inferred from a time-
dependent black hole movie with $\sigma=0.25$ (blue) or $\sigma=0.45$ (green)
and underlying parameters given in Table 1. We carry out $500$ fits across the
baseline window $[86,116]\,\text{G}\lambda$ using a visibility amplitude
produced by taking the coherent (solid lines) and incoherent (dashed lines)
averages of $N=10$ snapshots randomly chosen in each fitting attempt. The
dashed vertical lines represent the inferred diameters in the absence of
astrophysical fluctuations ($\sigma=0$). As $\sigma$ increases, the centers
and widths of the histograms do not undergo substantial changes, whereas their
areas decrease significantly: in the coherently-averaged case, $95\%$ of the
fits with $\sigma=0.25$ are successful, while that value decreases to $43\%$
when $\sigma=0.45$. For a small fraction of the randomly chosen sets of $N=10$
snapshots, we see the formation of a secondary peak corresponding to a
circlipse with the same fractional asymmetry as the one closest to the shape
of the critical curve, up to variations introduced by the instrument noise
(see Fig. 10).

Unlike the distributions of diameters inferred in the presence of instrument
noise (see Fig. 5), the diameters inferred in the presence of astrophysical
fluctuations now form a multi-peaked distribution (as shown in Fig. 9). As
discussed in the previous section, different realizations of instrument noise
always lead to a measurement of the _same_ circlipse as in the case
$s=0\,$mJy, up to a slight distortion and displacement. By contrast,
astrophysical fluctuations can change the structure of the image in ways that
instrument noise cannot. For instance, as the lower panel of Fig. 9 clearly
shows, astrophysical fluctuation can introduce photon ring mimickers that can
modify the periodicity in the visibility amplitude. The impact of this effect
on the inferred circlipse shape is exacerbated across windows located in
regions of the visibility domain that correspond to transitions between rings.

The circlipse corresponding to the smaller peak in Fig. 9 is the one directly
below the blue curve in the lower-left panel in Fig. 4, which shows a typical
fit with $\sigma=0.25$ in this survey. This second peak arises because the fit
is performed in a baseline window $[86,116]\,\text{G}\lambda$, which for this
particular profile lies within a transition region between the regimes
dominated by the first and second photon rings.

We recall that in the $s=0\,$mJy and $\sigma=0$ case, the inferred diameters
$d_{\varphi}$ jump between two circlipses (e.g., the circlipses starting at
$9.2M$ and $9.8M$ in Fig. 2). The upper-left panel of Fig. 4 shows that these
circlipses have similar joint goodness-of-fits in the absence of noise and
fluctuations: the joint goodness-of-fit of the “best” one (the blue curve in
Fig. 4) is $0.0023$, while that of the one directly below is $0.0016$. These
two values of the goodness-of-fit are “close” compared to the next highest
value of $0.00046$ (corresponding to the circlipse staring at $10.3M$), a
factor of $3.6$ times lower than the circlipse starting at $9.2M$.

Averaging over different sets of snapshots with non-zero astrophysical
fluctuations leads to changes in the visibility amplitude to which we apply
our fitting method. In a small fraction of our snapshot averages, the
visibility amplitude is such that this second circlipse has a larger goodness-
of-fit than the one closest to the shape of the critical curve, which leads us
to infer diameters that lie close to the smaller peaks in Fig. 9.

Interestingly, the green histogram in Fig 10 shows that the two circlipses
inferred in this survey have the same fractional asymmetry, up to slight
variations introduced by the fluctuations.

In other words, for a given geometry (i.e., for a fixed black hole spin and
observer inclination), the distribution of the inferred fractional asymmetry
is not multi-modal, while the inferred circlipses all differ from each other
by some constant scaling factor. As such, even though the diameters
$d_{\varphi}$ of the secondary circlipses are farther away from the shape of
the critical curve (whose overall size is set by the black hole mass), they
nonetheless retain some information about the spacetime geometry, encoded in
the ring asymmetry.

Figure 10: The distributions of the fractional asymmetry (17) inferred from
simulations of a black hole surrounded with the emission profile P1 specified
in Table 1, and with various levels of instrument noise $s$ and astrophysical
fluctuations $\sigma$. The vertical dashed line denotes the fractional
asymmetry of the “correct” circlipse inferred in the absence of either noise
or fluctuations (see Fig. 2). This figure includes the fractional asymmetry of
all of the circlipses inferred in the presence of astrophysical fluctuations
(see Figs. 9 and 11), suggesting that even if one infers a different circlipse
due to transition effects, nevertheless information about the fractional
asymmetry may still be recovered from such an observation.

Even when averaging over only $N=10$ snapshots, the distributions for the
inferred diameters $d_{\parallel}$ and $d_{\perp}$ are quite closely aligned
with the diameters inferred in the absence of astrophysical fluctuations, and
indeed both distributions display a similar width. That is, as $\sigma$ grows
from zero to $\sigma=0.25$, and subsequently to $\sigma=0.45$, we do not
observe a significant reduction in the accuracy or precision of the inferred
diameters. Rather, the most pronounced effect that we observe is a reduction
in the area of the histograms, or equivalently, an increase of the failure
rate for fitting attempts. The percentages of successful fits for
$\sigma=0.25$ and $\sigma=0.45$ are $95\%$ and $43\%$, respectively, with
similar figures for the incoherent averages.

Consistent with the effects of instrument noise, we see that in the presence
of increasingly large astrophysical fluctuations, our ability to detect a
periodic signal in the visibility amplitude is significantly impaired, while
the accuracy and precision of our fits, when we are able to obtain them, does
not decrease nearly as substantially.

When running these simulations, we first specify the geometrical parameters
$(a,\theta_{\rm o})$, from which we compute the critical curve analytically.
Thus, we know _a priori_ that the dominant peak in Fig. 9 is that of the
“correct” circlipse, that is, the one closest to the shape of the (in itself
unobservable) critical curve (see, e.g., Fig. 2).

The distance between the two inferred circlipses is approximately equal to
$1/u_{w}$, where $u_{w}$ represents the length of the baseline in the middle
of the window Paugnat  _et al._ (2022). Thus, in this scenario, the size
difference between the two circlipses is only a few percent. Therefore, in a
realistic observation, one could perhaps identify the “correct” peak provided
one had a strong prior on the mass-to-distance ratio, as is the case for Sgr
A*.

##  V Astrophysical fluctuations and instrument noise

Having considered the effects of instrument noise and astrophysical
fluctuations separately, we now study their combined effect on our ability to
infer an interferometric diameter. For each of the astrophysical profiles
listed in Table 1, we attempt $500$ fits across the baseline window
$[86,116]\,\text{G}\lambda$ after averaging over $N=5$, $10$, $20$, and $70$
snapshots generated from a movie with fluctuations of magnitude $\sigma=0.25$.

In each fit, we add a separate realization of instrument noise with
$s=1.5\,$mJy to each of the $N$ randomly chosen snapshots, and then take their
coherent and incoherent average to obtain a time-averaged visibility amplitude
to which we apply our fitting method. We present the resulting inferred-
diameter distributions for the coherent (solid histograms) and incoherent
(dashed histograms) averages in Fig. 11. The dashed vertical lines therein
represent the inferred diameters in the case of $s=0\,$mJy and $\sigma=0$, and
are identical to those in Figs. 5 and 9.

Figure 11: The distributions of diameters inferred from a time-dependent movie
of a black hole surrounded with the emission profile P1 (upper panel) or P2
(lower panel) and underlying parameters given in Table 1, in the presence of a
level $s=1.5\,$mJy of instrument noise and $\sigma=0.25$ of astrophysical
fluctuations. We carry out $500$ fits across the baseline window
$[86,116]\,\text{G}\lambda$ using a visibility amplitude produced by taking
the coherent (solid lines) and incoherent (dashed lines) averages of $N=5$
(purple), $N=10$ (green), $N=20$ (blue), and $N=70$ (red) snapshots randomly
chosen in each fitting attempt. The dashed vertical lines represent the
diameters inferred in the absence of instrument noise and astrophysical
fluctuations (i.e., $s=0\,$mJy and $\sigma=0$). For the profile P1, the
baseline window lies (at every angle $\varphi$) within a transition region
between the regimes dominated by the first and second photon rings. As a
result, the distributions are multi-peaked and, for $N=70$, of roughly the
same height. For the profile P2, which lies in the regime dominated by the
$n=1$ ring for most angles $\varphi$, the distributions effectively have a
single peak that corresponds to the interferometric diameter
$d_{\varphi}^{(1)}$ of the ring. For both profiles, as the number $N$ of
snapshots increases, the histograms become narrower and the percentage of
successful fits increases significantly.

For the profile P1, the inferred-diameter distributions in Fig. 11 are multi-
peaked, with two peaks of similar heights separated by $\Delta d_{\phi}\approx
1/u_{w}$, where $u_{w}$ is the length of the baseline in the middle of the
sampled window. This suggests that in the presence of fluctuations, the
addition of instrument noise exacerbates any transition effects and leads to
the inference of a wrong circlipse almost equally as it does the correct one.
By contrast, for the profile P2, the visibility amplitude is not always
sampled in a transition region at each of the $36$ baseline angles (see, e.g.,
the upper panel of Fig. 4 in Ref. Cárdenas-Avendaño and Lupsasca (2023)). As a
result, the inferred-diameter distributions for profile P2 have effectively
only one peak. Indeed, Fig. 11 shows that already with $N=20$ snapshots, one
always infers the same, correct circlipse.

This suggests that in an actual observation, if one were to obtain two such
peaks by randomly sampling different subsets of the measured data, then one
might be led to conclude that the visibility amplitude was measured in a
transition region. But even if this happens, and one does not have a prior
that is sufficiently strong to break the degeneracy, Fig. 10 suggests that the
measured diameter $d_{\varphi}$ still retains correct information about the
fractional asymmetry of the photon ring.

For the profile P1, Fig. 11 shows that averaging has a very strong effect,
even with only $N=5$ snapshots: the peaks of the (purple) $N=5$ histograms
corresponding to the correctly inferred circlipse are more closely aligned
with the dashed vertical lines than are the $s=1.5\,$mJy histograms in Fig. 5,
even if astrophysical fluctuations are now present. For $\sigma=0.45$,
however, the distributions of diameters inferred from $N=5$ snapshots are
quite wide, indicating that more snapshots are required to tame this level of
fluctuation.

As the scale of astrophysical fluctuations and the level of instrument noise
grow larger, the dominant effect for profile P1 is a reduction in the areas of
the histograms. The $N=5$ histograms for $\sigma=0.25$ have much smaller areas
than the $s=1.5\,$mJy histograms in Fig. 5. More precisely, in the presence of
only instrument noise with a level $s=1.5\,$mJy, $95\%$ of the fits are
successful, but this figure decreases to $76\%$ (or $26\%$) after the addition
of astrophysical fluctuations with $\sigma=0.25$ (or $\sigma=0.45$) and after
averaging over only $N=5$ snapshots. These figures, however, increase
substantially as $N$ increases: for $N=70$ snapshots, $100\%$ of the fits with
$\sigma=0.25$ are successful, and likewise, out of the $500$ fitting attempts
for $\sigma=0.45$, only one fails. However, for the profile P2, even with only
$N=5$ snapshots, $99\%$ (or $83\%$) of the fits are successful in the presence
of fluctuations at a scale $\sigma=0.25$ (or $\sigma=0.45$): as expected, it
is easier to detect a periodicity outside of a transition region, and this is
in fact possible even with as few as $N=5$ snapshots.

In summary, the combination of instrument noise and astrophysical fluctuations
leads to two main effects: they make it harder to detect a periodicity in the
visibility amplitude, and they exacerbate transition effects. When observing
in a transition region, the inferred-diameter distributions may become double-
peaked. This results in an ambiguity in the overall scale of the angle-
dependent ring diameter $d_{\varphi}$, but fortuitously, not in its fractional
asymmetry. Outside of a transition region, the inferred-diameter distributions
effectively have only one peak. In either case, averaging over snapshots is a
powerful tool: as $N$ increases, the periodicity of the average visibility
amplitude becomes increasingly pronounced, even when both astrophysical
fluctuations and instrument noise are present, and the inferred-diameter
distributions become increasingly narrow. In other words, our survey shows
that, using only our simple data analysis method, it is possible to measure
the shape of the photon ring quite accurately and precisely, even though
transitions effects may result in a loss of information about the overall ring
scale, and thus make parameter inference more difficult.

##  VI Conclusions

We have conducted a systematic study to investigate how instrument noise and
astrophysical fluctuations can impact our ability to infer the shape of the
photon ring. Our findings reveal that it is feasible to detect a periodic
signal in the visibility amplitude up to a certain threshold level of noise
and astrophysical fluctuations.

Remarkably, up to this point, neither the instrument noise nor the
fluctuations significantly compromise the accuracy or precision of our
simulated measurements. However, our analysis also identifies a critical
threshold—a sharp transition—beyond which such a measurement becomes
intractable with our fitting method.

Measuring an interferometric diameter and comparing it to the GR prediction is
challenging in the presence of both instrument noise and source fluctuations.
However, time-averaging over several snapshots is a potent means for restoring
the feasibility of a photon ring measurement. Although our analysis
underscores the formidable impact of instrument noise and astrophysical
fluctuations, we do not rule out the possibility that a more sophisticated
data analysis scheme could still detect a periodic signal in the visibility
amplitude, even when our method could not. Thus, further studies of data
analysis techniques may present avenues for overcoming the limitations imposed
by instrument noise and help refine our understanding of future observations.

It is especially challenging to infer an interferometric diameter from
observations of the visibility amplitude in a baseline window where the signal
of the first photon ring does not always dominate. For example, sampling the
visibility in a transition region between two regimes dominated by different
rings can sometimes result in the inference of another circlipse. Such a
secondary circlipse may differ from the one whose shape is closest to that of
the critical curve by an overall scaling factor, but nonetheless retains the
same fractional asymmetry (up to noise effects). In other words, even when
observations are performed in a transition region, it remains possible to
measure a photon ring shape with the correct fractional asymmetry, even if the
inferred diameter is in some cases off by a constant. Such effects can be
mitigated using various consistency checks. For instance, when analyzing a
sequence of images of a black hole, one would expect consistent results across
different sets of snapshots, which should lead to the inference of the same
circlipse shape or consistent diameters at every baseline angle. However,
should discrepancies arise, they could serve as potential indicators that the
observations have been performed in a transition region, and hence that their
analysis requires a more sophisticated approach.

This first study has only scratched the surface of the difficulties inherent
in measurements of the photon ring shape amidst noise and fluctuations. This
research is still in its early stages, and there are many ways to improve and
refine our approach.

First, we have relied on a particular statistical method for modeling
astrophysical fluctuations. This approach has allowed us to simulate various
emission models in a less computationally expensive way compared to running
GRMHD simulations. The fluctuations in our models all exhibit a correlation
structure governed by the highly versatile Matérn covariance function, which
enjoys broad applicability and can even describe free scalar fields in
Euclidean field theory Cárdenas-Avendaño  _et al._ (2023).

By assessing whether the Matérn covariance function accurately reproduces the
correlation structure observed in GRMHD simulations, we not only probe the
intricacies of turbulence in accretion flows but also strive to bridge the gap
between mathematical models and real physical systems. Varying this covariance
is an exciting prospect for future exploration.

Our rather simplistic approach for the incorporation of instrument noise in
simulated observations—consisting of the addition of Gaussian complex
noise—paves the way for future analyses that will consider additional factors.

Modeling detailed features of future VLBI observations with a radio dish in
space (such as its size and orbit), along with real-world conditions (such as
weather) and a realistic array of multiple ground-based stations promises to
refine our understanding of observational constraints. Some of these aspects
have been recently studied using GRMHD simulations and several possible
configurations for an orbiting satellite in Ref. Shlentsova  _et al._ (2024),
which also found that space-VLBI missions can be used to image the variability
in a black hole environment and its extended jets.

Currently planned space missions will target M87∗ and the supermassive black
hole at the center of our galaxy, Sgr A*. Observations of Sgr A* present more
challenges posed by the propagation of the emitted light through the
interstellar medium, and by the shorter timescale for variability in the
source. In particular, phenomena such as scattering cannot be neglected, which
necessitates a reassessment of our methodologies and assumptions.

This work is a proof of concept for critical components in the data analysis
pipeline that are essential to future missions targeting black hole photon
rings. Our results are part of the ongoing effort to transform black hole
images into precision probes of fundamental phenomena in astrophysics, and
offer encouraging prospects for future measurements of black hole parameters
via photon ring observations.

###### Acknowledgements.

We thank George Wong for many helpful discussions. We are grateful to William
and Kacie Snellings for their generous support. A.C.-A. also acknowledges
support from the Simons Foundation, and A.L. is supported in part by NSF
grants AST-2307888 and PHY-2340457. The simulations presented in this article
were performed on computational resources managed and supported by Princeton
Research Computing, a consortium of groups that includes the Princeton
Institute for Computational Science and Engineering (PICSciE), and the Office
of Information Technology’s High Performance Computing Center and
Visualization Laboratory at Princeton University.

##  Appendix A Fits across the RMSD threshold

In Sec. III.3, we chose a threshold RMSD of $0.05\%$ to draw as a white
contour in Fig. 7. In our survey, we find that an RMSD of $0.05\%$ provides a
conservative value for our simulated observations. To illustrate this, we pick
a particular profile (black hole spin $a/M=94\%$, observer inclination
$\theta_{\rm o}=10^{\circ}$, and emission profile parameters $\mu=r_{+}$,
$\vartheta=M$, and $\gamma=0$) and perform fits across two arbitrarily chosen
baseline windows, $[76,96]\,\text{G}\lambda$ and $[76,101]\,\text{G}\lambda$,
which are enclosed by the orange and red borders in Fig. 7, respectively, and
lie on either side of the threshold RMSD region.

The minimum and average power (14) across the first window are $1.8\,$mJy and
$2.3\,$mJy, respectively, and $2.0\,$mJy and $2.4\,$mJy across the second
window. In Fig. 12, we show the best-fit circlipses obtained from a single fit
across these baseline windows in the absence of noise, i.e., $s=0\,$mJy
(dashed lines), and then in the presence of instrument noise at a level
$s=1.5\,$mJy (solid lines). The RMSD of the fits across
$[76,96]\,\text{G}\lambda$ and $[76,101]\,\text{G}\lambda$ are $0.058\%$ and
$0.048\%$, respectively. The fit across $[76,96]\,\text{G}\lambda$ is markedly
worse than the one across $[76,101]\,\text{G}\lambda$, with the inferred
diameters for the former deviating more from its circlipse than the latter.
Indeed, the best-fit circlipses inferred in the presence of noise (solid
lines) exhibit more significant deviations from their corresponding noiseless
analogues for the fit across $[76,96]\,\text{G}\lambda$ (dashed purple line)
than for the fit across $[76,101]\,\text{G}\lambda$ (green solid line).

Figure 12: Inferred photon ring shapes (6) for a black hole with spin
$a/M=94\%$ surrounded by a radial emission profile (2) with parameters
$\mu=r_{+}$, $\vartheta=M$, and $\gamma=0$, and observed from an inclination
$\theta_{\rm o}=10^{\circ}$, in the presence of instrument noise levels
$s=0\,$mJy (dashed lines) and $s=1.5\,$mJy (solid lines), across the baseline
windows $[76,96]\,\text{G}\lambda$ (purple crosses) and
$[76,101]\,\text{G}\lambda$ (green solid points). On these baselines, the
visibility amplitude attains a minimum power (14) of $1.8\,$mJy and
$2.3\,$mJy, and an average power of $2.0\,$mJy and $2.4\,$mJy, respectively.
The RMSDs for these fits are $0.058\%$ (purple solid lines) and $0.048\%$
(green solid line), so they lie just above and below the critical RMSD of
$0.05\%$, respectively.

As an indication of the quality of fit for this profile, for the fit across
$[76,101]\,\text{G}\lambda$, the percentage change in the inferred diameters
$d_{\perp}$ and $d_{\parallel}$ when going from $s=0\,$mJy to $s=1.5\,$mJy is
$0.044\%$ and $0.035\%$, respectively, while the inferred diameters of the
first ring differ from those of the critical curve by $0.40\%$ and $0.28\%$,
respectively.

##  Appendix B Power in the baseline window $[86,116]\,\text{G}\lambda$ across
all models

To characterize the strength of the visibility amplitude across some given
baseline window and at some baseline angle $\varphi$, we show in Fig. 13 the
distribution of the power (14) across the baseline window
$[86,116]\,\text{G}\lambda$ for all the profiles listed in Tables 2 and 3. The
notion of power is especially important when thinking about the physical
effects of some noise level, since, in combination with $s$, it provides a
rough estimate of the SNR.

From Fig. 13, one can see that the instrument noise levels $s=1.5\,$mJy and
$s=2.5\,$mJy considered in the main text should indeed have a significant
effect on the underlying visibility signal (as the right panels of Fig. 4 also
show).

Figure 13: Distribution of the minimum (solid histrograms) and average (dashed
histrograms) power (14) across the baseline window $[86,116]\,\text{G}\lambda$
for the $270$ time-averaged profiles (pink) given in Table 2, and for the
$108$ time-dependent profiles with astrophysical fluctuations (light blue)
given in Table 3. For the time-averaged profiles, the minimum power attained
at any baseline angle is $0.2\,$mJy, while the average power across all the
baselines windows and all the profiles is $3.7\,$mJy. For the time-dependent
profiles, the minimum and average powers are $0.03\,$mJy and $2.2\,$mJy,
respectively.

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