11institutetext: Institut für Theoretische Physik, Goethe Universität, Max-
von-Laue-Str. 1, D-60438 Frankfurt, Germany 22institutetext: Black Hole
Initiative at Harvard University, 20 Garden Street, Cambridge, MA 02138, USA
33institutetext: Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69,
D-53121 Bonn, Germany 44institutetext: Tsung-Dao Lee Institute and School of
Physics and Astronomy, Shanghai Jiao Tong University, Shanghai, 200240,
People’s Republic of China 55institutetext: Mullard Space Science Laboratory,
University College London, Holmbury St. Mary, Dorking, Surrey RH5 6NT, UK
66institutetext: Department of Astrophysics/IMAPP, Radboud University
Nijmegen, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands 77institutetext:
Anton Pannekoek Institute for Astronomy, University of Amsterdam, Science Park
904, 1098 XH Amsterdam, The Netherlands 88institutetext: Dipartimento di
Fisica “E. Pancini”, Universitá di Napoli “Federico II”, Via Cinthia, I-80126,
Napoli, Italy 99institutetext: INFN Sez. di Napoli, Via Cinthia, I-80126,
Napoli, Italy 1010institutetext: Jodrell Bank Centre for Astrophysics,
University of Manchester, Machester M13 9PL, UK 1111institutetext: Frankfurt
Institute for Advanced Studies, Ruth-Moufang-Strasse 1, 60438 Frankfurt,
Germany 1212institutetext: School of Mathematics, Trinity College, Dublin 2,
Ireland
1212email<EMAIL_ADDRESS>
# Using space-VLBI to probe gravity around Sgr A∗

C. M. Fromm 112233 Y. Mizuno 4411 Z. Younsi 5511 H. Olivares 6611 O. Porth
7711 M. De Laurentis 889911 H. Falcke 6633 M. Kramer 331010 L. Rezzolla
1111111212

( Draft 1.0: )

###### Abstract

Context. The Event Horizon Telescope (EHT) will soon provide the first high-
resolution images of the Galactic Centre supermassive black hole (SMBH)
candidate Sagittarius A* (Sgr A∗), enabling us to probe gravity in the strong-
field regime. Besides studying the accretion process in extreme environments,
the obtained data and reconstructed images could be used to investigate the
underlying spacetime structure. In its current configuration, the EHT is able
to distinguish between a rotating Kerr black hole and a horizon-less object
like a boson star. Future developments can increase the ability of the EHT to
tell different spacetimes apart.

Aims.  We investigate the capability of an advanced EHT concept, including an
orbiting space antenna, to image and distinguish different spacetimes around
Sgr A∗.

Methods. We use general-relativistic magneto-hydrodynamical (GRMHD)
simulations of accreting compact objects (Kerr and dilaton black holes, as
well as boson stars) and compute their radiative signatures via general-
relativistic radiative transfer (GRRT). To facilitate comparison with upcoming
and future EHT observations we produce realistic synthetic data including the
source variability, diffractive and refractive scattering while incorporating
the observing array, including a space antenna. From the generated synthetic
observations we dynamically reconstructed black hole shadow images using
regularised Maximum Entropy methods. We employ a genetic algorithm to optimise
the orbit of the space antenna with respect to improved imaging capabilities
and u-v-plane coverage of the combined array (ground array and space antenna
and developed a new method to probe the source variability in Fourier space.

Results. The inclusion of an orbiting space antenna improves the capability of
the EHT to distinguish the spin of Kerr black holes and dilaton black holes
based on reconstructed radio images and complex visibilities.

###### Key Words.:

Physical data and processes: Gravitation, Magnetohydrodynamics (MHD),
radiation mechanisms: thermal – Methods: numerical – Galaxies: individual: Sgr
A∗ – Techniques: interferometric

##  1 Introduction

The Event Horizon Telescope presented the first horizon scale images of the
black hole in M87 (Event Horizon Telescope Collaboration et al., 2019a) and
will soon provide the first image of the black hole candidate Sgr A∗ in our
Galaxy. The current configuration of the EHT consists of eight telescope
scattered across Europe, North- and South America and the South Pole (Event
Horizon Telescope Collaboration et al., 2019b). Combining the data recorded
simultaneously by the individual telescopes after the observations, $\mu$as
angular resolution is achieved. These observations enable us, for the first
time, to study accretion processes in M87 and in the centre of our galaxy with
unparalleled resolution and to probe gravity in the strong-field regime. Given
the small number of telescopes participating in the observations, the
intrinsic variability of the source and the interstellar scattering screen,
reconstructing an image and discriminating among different spacetimes is
challenging (Lu et al., 2014, 2016; Mizuno et al., 2018). An interferometer
such as the EHT samples the brightness distribution of an astronomical object
in Fourier-space: the so called _u-v plane_. Due the limited number of
participating telescopes, the u-v plane is sparsely sampled. The sampling of
this u-v plane increases with the number of telescopes and the duration of the
observations. However, an improvement in the resolution of the image can be
only be obtained by increasing the distance between the telescopes, the so-
called baselines, or by increasing the observed frequency. Given the current
configuration of the EHT array, a significant increase in the baselines can
only be achieved by extending baselines to space. Using the ground array while
increasing the observed frequency above 345 GHz is limited by the opacity of
the atmosphere and its water vapour content.

The concept of space-based Very Long Baseline Interferometry (VLBI) including
a ground array has been studied extensively since the early 1970s (see
Schilizzi, 2012, for a historic overview) and successfully launched space
antennas are Highly Advanced Laboratory for Communications and Astronomy
(HALCA) (project VLBI Space Observatory Programme, (VSOP)) (Hirabayashi et
al., 1998) and, more recently, Spektr-R (project RadioAstron) (Kardashev et
al., 2013). Both missions operate at lower frequencies than the EHT (230 GHz)
and in the case of RadioaAstron provide $\rm\mu as$ resolution (Gómez et al.,
2016). Therefore, in this work we consider the increase in the angular
resolution that can be obtained by extending the baselines of the EHT via a
space-based antenna (see also Palumbo et al., 2019). This configuration has
the advantage of using a well-calibrated ground array. For a mission based
entirely on space telescopes, see a recent studies by Roelofs et al. (2019);
Fish et al. (2019). Within this work we address the scientific question as to
whether such a configuration will improve the current ability of the EHT to
distinguish among different theories of gravity using radio observations of
Sgr A∗.
The structure of this paper is as follows: in Sect. 2 we briefly introduce our
GRMHD simulations and GRRT calculations. The procedure for the selection of
the orbit of the space antenna is introduced in Sect. 3 and the results of the
synthetic imaging and data analysis are shown in Sect. 4. We present our
discussion and conclusions in Sect. 5. Sgr A∗ is located at RA:
$17^{h}\,45^{m}\,40.0409^{s}$ and
DEC:$-29^{\circ}\,45^{\prime}\,40.0409^{\prime\prime}$, at a distance of $\rm
8178\pm 13\,stat.\pm 22\,sys.\,pc$ and its central candidate SMBH exhibits a
mass of $M_{\rm bh}=4.14\times 10^{6}M_{\astrosun}$ (Gravity Collaboration et
al., 2019).

##  2 GRMHD and GRRT simulations

We use the state-of-the-art GRMHD code BHAC (Porth et al., 2017; Olivares et
al., 2019) and perform three-dimensional accretion simulations on to Kerr
black holes (${\rm a}_{*}$=0.6 and ${\rm a}_{*}$=0.94), dilaton black holes
and a boson star (Mizuno et al., 2018; Olivares et al., 2020). Here ${\rm
a}_{*}$ is the dimensionless black-hole spin parameter, with $|\rm a_{*}|<1$.

Figure 1: Top: panels A and B show GRRT images for a Kerr black hole with
${\rm a}_{*}$=0.6 including scattering effects ( diffractive and refractive
scattering) for two selected times t=12.6 h and t=23.6 h (dash-dotted line in
the bottom panel) at a frequency of 230 GHz. The average image for 24 hours of
observation is presented in panel C. Bottom: simulated total flux light curve
at 230 GHz for 24 hours of observations. Indicated is a typical EHT
observation with 12 hours duration.

We have chosen a black hole solution in the Einstein-Maxwell-dilaton-axion
(EMDA) theory, which derives from a particular string theory (Okai, 1994;
García et al., 1995). Such black-hole solutions exhibit more diverse physical
properties than others in the literature. For example, they have hair, in
addition to mass, rotation and charge, so as to be an interesting laboratory
for performing black hole experiments and studying possible differences
between Einstein’s gravity and alternative theories of gravity. Moreover,
since both the dilaton and the axion are considered to be candidates for dark
matter, the study of the shadow from a dilaton black hole may provide hints on
the observational properties of the matter in which the black hole is immersed
(Davoudiasl & Denton, 2019).

We also consider a stable boson star, which although classified as a compact
object, is physically completely different from a black hole (Schunck &
Liddle, 1997). It is well known that the gravitational field of a boson star
bends light around itself, creating a region _resembling_ the shadow of a
black hole’s event horizon. Like a black hole, a boson star will accrete
ordinary matter from its surroundings, but its opacity means this matter
(which likely would heat up and emit radiation) would be visible at its
center. There is no significant evidence so far that such stars exist.
However, it may become possible to detect them through VLBI measurements
(Olivares et al., 2020). In theory, a super-massive boson star could exist at
the core of a galaxy, which might explain many of the observed properties of
active galactic nuclei. Boson stars have also been proposed as candidate dark
matter objects, and it has even been hypothesised that the dark matter haloes
surrounding most galaxies might be viewed as enormous boson stars (Levkov et
al., 2018).

The formation of boson stars or other exotic objects is an interesting process
to study. Although we know that at the center of our galaxy there is a highly
compact object, it is important to study different candidates because we might
find that other galaxies could harbour exotic objects composed of “non-
baryonic” matter at their centres (Lee & Koh, 1996).

For spacetimes not described by Einstein’s general theory of relativity, we
use the Rezzolla-Zhidenko metric parameterisation (Rezzolla & Zhidenko, 2014;
Younsi et al., 2016). The initial conditions of all GRMHD simulations for Kerr
and dilaton black holes as well as for the boson star, consist of a torus in
hydrodynamical equilibrium with a weak poloidal magnetic field. In order to
trigger the accretion process the magnetic field in the torus is seeded with a
small perturbation which leads to the formation of the magneto-rotational
instability (MRI). After the saturation of the MRI (${\rm
t}>5000\,\mathrm{GM/c^{2}}$, with gravitational constant, G, mass of the
central object, M, and speed of light c) all simulations show a quasi-
stationary accretion flow (for more details see Mizuno et al., 2018; Olivares
et al., 2020).

In the next step we compute the radiative signature of the accretion process
via GRRT calculations using the BHOSS code (Younsi et al., 2020; Gold et al.,
2020). We use an viewing angle of $\theta=60^{\circ}$, adjust the ion-to-
electron temperature ratio ($T_{i}/T_{e}=3$) and the mass-accretion rate to in
order to obtain the an average flux of 4 Jy at 230 GHz (see supplemental
information in Mizuno et al., 2018, for a detailed description of the GRRT
calculations and fitting procedure). This leads to a flux density variation
between 3 Jy and 5 Jy (see light curve in the bottom panel of Fig. reffluxvar)
which is in agreement with the observed flux density variations 2 Jy – 5.5 Jy
provided by Bower et al. (2015) The GRRT images are computed every 10 M which
corresponds to 200 s for SgrA* for a time span of 4320 M (24 h for SgrA). This
long duration allows for the calculation of several overlapping and two
independent 12 h111typical duration for EHT observations of SgrA∗ which allows
European and American telescopes to participate in the observation.
observational window. In this work we investigate the following spacetimes:
Kerr black holes in general relativity, a dilaton black hole and a boson star.
The variability of the total flux and in the emission structure of the
individual GRRT snapshots, together with the averaged image, is presented in
Fig. 1 and in Table LABEL:GRRTinfo we present an overview of the GRRT images
used.

Table 1: Overview of simulations used Spacetime | ${\rm a}_{*}$ |  $\theta$ [deg] |  $\Delta t$ [M (s)] | t [M (h)]
---|---|---|---|---
Kerr | 0.60 | 60 | 10 (200) | 4320 (24)
Kerr | 0.94 | 60 | 10 (200) | 4320 (24)
dilaton | 0 | 60 | 10 (200) | 4320 (24)
boson star | 0 | 60 | 10 (200) | 4320 (24)

##  3 Orbit selection and optimisation strategy

In this section of our exploratory paper we present one possible array
optimisation strategy which could lead to improved imaging capabilities for
the EHT and thus may allow us to distinguish among different spacetimes. Thus,
the question arises as to which kind of orbit is required for improving the
imaging of the Galactic Centre. The answer to this question depends strongly
on the observational constraints: the position of the Galactic Centre relative
to the space and ground antennas and the duration of the observations. In the
following we will assume an observation schedule consisting of 12 hours of Sgr
A∗ observations. This long observing time allows European as well as North-
and South American telescopes to participate in the observation.

For the space-EHT concept we consider the EHT 2017 configuration as the ground
array and include the Northern Extended Millimeter Array (NOEMA222http://iram-
institute.org/EN/) and a space antenna. In Table 2 we list each antenna and
its corresponding system equivalent flux density (SEFD) (Event Horizon
Telescope Collaboration et al., 2019b).

Table 2: Effective antenna diameter $d$ (for single telescopes this corresponds to the diameter of the dish) and system equivalent flux density (SEFD) used for the ground and space antennas (see Event Horizon Telescope Collaboration et al., 2019b, for details) Telescope |  $d_{\mathrm{eff}}$ [m] | SEFD [Jy]
---|---|---
ALMAa | 73 | 74
APEX | 12 | 4700
JCMT | 15 | 10500
LMT | 32.5 | 4500
NOEMA∗ | 52 | 700
PV | 30 | 1900
SMT | 10 | 17100
SMAb | 14.7 | 6200
SPTc | 6 | 19300
Space Antenna∗ | 8 | 20000
∗ only used for the space-EHT configuration.
a for EHT 2017 ALMA used 37 $\times$ 12 m.
b for EHT 2017 SMA used 6 $\times$ 6 m.
c for EHT 2017 SPT was under-illuminated
with an effective diameter of 6 m.

For the orbiting space antenna we assume a diameter of 8 meters (similar to
the size of Spektr-R), a system temperature of 100 K and an antenna efficiency
of 40%. Given these values, a SEFD of $20\times 10^{3}$ Jy is computed. The
orbit of the space antenna can be described by six orbital elements which can
be divided into orbital shape parameters (semi-major axis, $a$, and
eccentricity, $e$) and orbital orientation parameters which provide the
location and orientation of the orbit in space relative to the equatorial
plane (inclination, $i$, longitude of the ascending node, $\Omega$, argument
of perigee, $\omega$, and the true anomaly, $\vartheta$). The six orbital
elements of the space antenna together with duration of the observations lead
to a seven-dimensional parameter space which has to be searched to obtain an
optimal configuration. This task can be formulated as a constrained non-linear
optimisation problem and written as follows:

$\displaystyle\begin{array}[]{ll@{}ll}\mathrm{minimize:}&\displaystyle
f(\@vec{x})\,,&\\\ \mathrm{subject\ to:}&\displaystyle g_{j}(\@vec{x})\leq
0\,,&&j=1,...,n\,,\\\ &\displaystyle x_{L,i}\leq x_{i}\leq
x_{R,i}\,,&&i=1,...,m\,,\end{array}$ (4)

where $\@vec{x}$ is a seven-dimensional vector containing the model
parameters, i.e.,
$\@vec{x}\equiv\left[t_{\mathrm{obs}},a,e,i,\Omega,\omega,\vartheta\right]^{T}$,
$f(\@vec{x})$ is the objective function (minimisation function),
$g_{j}(\@vec{x})$, are the constraints and $x_{L,i}$ and $x_{R,i}$ are lower
and upper boundaries for the model parameters. In Table LABEL:parabound we
report the boundaries used during the optimisation.

Table 3: Parameter boundaries used for the optimisation $t_{\rm obs}$ [h] |  $a$ [$10^{3}$km] | $e$ |  $i$ [deg] |  $\Omega$ [deg] |  $\omega$ [deg] |  $\vartheta$ [deg]
---|---|---|---|---|---|---
0 -12 | 0.4-100 | 0-0.9 | 0-180 | 0-360 | 0-360 | 0-360

Improving the imaging capabilities of the EHT can be translated into an
increased angular resolution and a denser sampling of the u-v plane as
compared to the current EHT configuration. The improvement of the imaging
capabilities i.e., better array resolution and u-v plane coverage can be
assessed by computing image metrics between the infinite resolution GRRT image
and the reconstructed image. Therefore, we use for the minimisation process a
combination of the denser sampling in the u-v plane (minimising the distance
between u-v points) and the improved imaging (improved image metrics). For the
image metrics we used the normalised cross correlation coefficient (nCCC) and
the structural dissimilarity measure (DSSIM)333smaller values indicate a
better image agreement (Wang et al., 2004). The minimisation function is given
by:

$f\left(\@vec{x}\right)=\mathrm{DSSIM}\left(\@vec{x}\right),$ (5)

and we use the following constraints444the constraints are fulfilled if the
$g_{i}<0$ to speed up the optimisation procedure:

$\displaystyle g_{1}$ $\displaystyle=$
$\displaystyle\mathrm{DSSIM}\left(\@vec{x}\right)-0.75\times\mathrm{DSSIM_{EHT2017}},$
(6) $\displaystyle g_{2}$ $\displaystyle=$ $\displaystyle\Delta r_{\rm
uv,\,max}\left(\@vec{x}\right)-\Delta r_{\rm uv,\,max}^{\rm EHT2017}\,.$ (7)

The first constraint ensures that the DSSIM is improved by at least 25% as
compared to image reconstructed using the EHT 2017 configuration. The denser
sampling of the u-v plane is addressed by the second constraint, $g_{2}$,
where we enforce that the projected distance between the u-v points, $\Delta
r$, is minimised with respect to u-v sampling of the EHT 2017 configuration.
During the optimisation we generate for each set of parameters e.g., for each
$\left(\@vec{x}\right)$ synthetic visibilities including both, diffractive and
refractive scattering (Johnson, 2016) and reconstruct the image using the
EHTim package 555https://github.com/achael/eht-imaging (Chael et al., 2016,
2018).

Due to the numerical costs we use during the optimisation a single GRRT image
instead of a series of GRRT images (or GRRT movie) and the array configuration
and additional parameters used during the data generation and image
reconstruction are listed in Table 2 and 4. The procedure of the image
reconstruction follows the approach of Chael et al. (2016): initialisation of
the imaging using as gaussian prior with a FWHM of 70 $\mu$as and the repeated
re-initialisation of the imaging using the previously obtained image convolved
with half the nominal array resolution. During the orbit optimisation we use
two re-initialisation loops and we align the reconstructed image with the GRRT
image prior to the computation of the DSSIM.

Table 4: Parameters used for the data generation and image reconstruction
synthetic data generation
---
$t_{\rm int}$ [s] |  $\Delta\nu$ [GHz] |  $t_{\rm gap}^{a}$ [s] |  $\nu_{\rm obs}$ [GHz] | gain off setb
12 | 4, 8∗ | 200 | 230 | 0.1
image reconstruction
data | weightingc | regularizer | weightingc
visibilities | 100 | simple entropy | 2
bi-spectra | 10 | simple entropy | 1
∗ only used for the space-EHT configuration.
a time difference between GRRT images
b antenna gains are drawn form a normal distribution with
mean=1.0 and standard deviation=0.1 (gain off set).
c weighting factor for data terms and image entropies
used during the image reconstruction.

Figure 2: Result of the orbit optimisation for 12 hours of Sgr A∗
observations. Top row from left to right: GRRT image (panel a), reconstructed
image with interstellar scattering (including both, diffractive and refractive
scattering during the generation of the synthetic visibilities) using the EHT
2017 configuration convolved with 75% (red shading) of the nominal array
resolution (light grey shading, panel b) and reconstructed image with
interstellar scattering (including both, diffractive and refractive scattering
during the generation of the synthetic visibilities) using the space-EHT
configuration convolved with 75% (red shading) of the nominal array resolution
(light grey shading, panel c) for a dilaton black hole. Bottom row from left
to right: satellite orbit as seen from SgrA⋆ with orbital parameters and
orbital period (panel d), satellite ground track (red lines for 12 hours and
grey ones for 24 hours) and ground array antennas (blue points, panel e) and
u-v sampling for the ground array (white points) and the baselines including
the space antenna (red points, panel f).

Given the high dimensionality of this constrained optimisation problem,
gradient-based solvers may become stuck in a local minimum and/or required
large computational resources to map out the gradient of the parameter space
with sufficient resolution to avoid this problem. An elegant method to
circumvent the above mentioned difficulties is based on gradient-free
optimisation algorithms. In this work we apply a genetic algorithm (GA) and in
particular employ the implementation of a Non Sorting Genetic Algorithm II
(NSGA2) to solve the optimisation problem (Deb et al., 2002). For the initial
generation we use 1000 randomly initiated orbits and we evolve them for 100
generations.

The results of our numerical optimisation are presented in Table 5 and Fig. 2.
The improvement of the image reconstruction is clearly visible by comparing
the EHT 2017 reconstructed image (panel b in Fig. 2) with the image obtained
by space-EHT (panel c in Fig. 2). The ground truth GRRT image is presented in
panel a in Fig. 2. The space-EHT image is able to capture and image fine flux
arcs which are smeared out in the EHT 2017 configuration. The visible image
improvements are also reflected in the improved image metrics: the nCCC
increased from 0.92 to 0.96 and the DSSIM decreased by 40% from 0.252 to 0.152
(see also constraint $g_{1}$ Eq. 7). Notice that increased nCCC and decreased
DSSIM corresponds to a better image matching. The satellite orbit as seen from
Sgr A⋆ (note that the Earth is viewed from $-30^{\circ}$) is presented in
panel d in Fig. 2. In panel e in Fig. 2 the satellite ground track (projection
of the satellite orbit onto the surface of the Earth) of the space antenna
(red points for 12 hours and grey ones for 24 hours) and the ground antennas
are indicated by the blue points. The sampling of the u-v plane is presented
in panel f in Fig. 2 where the white points indicate the u-v tracks of the
ground array and the red points the u-v tracks including the space antenna.
The addition of the space antenna is not only extending the u-v sampling up to
25 G$\lambda$ but also adds short and intermediate baselines to the array as
required by the constraint $g_{2}$ (see Eq. 7) of the optimisation process
which also improve the imaging capabilities.

Table 5: Optimised satellite orbit for 12 hours of Sgr A∗ observation. a [km] | i [∘] | e |  $\Omega$ [∘] |  $\omega$ [∘] |  $\vartheta$ [∘] |  $t_{\mathrm{obs,start/stop}}$ [UT] |  $\mathbf{T_{p}}$ [h]
---|---|---|---|---|---|---|---
14900 | 67 | 0.5 | 46 | 70 | 330 | 04:00 – 16:00 | 5.03

The orbit optimisation procedure suggested an elliptical orbit with a semi-
major axis of $a=14900$ km with an eccentricity of $e=0.5$ at an inclination
of 67∘. The optimal time span of the SgrA⋆ observation is obtained between
4:00 UT and 16:00 UT. The orbital period of the satellite is $\rm T_{p}=5.03$
h. Notice, that elliptical orbits with long orbital periods have also been
used for VSOP ($e$=0.6, $\rm T_{p}=6.3$ h) and the Spekt-R ($e=0.9$, $\rm
T_{p}$=200 h) (Hirabayashi et al., 1998; Kardashev et al., 2013). In Appendix
A we illustrate the impact of different satellite orbits on the image
reconstruction. Our results differ from the results of Palumbo et al. (2019)
which used a circular orbit with semi-major axis $a=6652$ km at an inclination
of 61∘. The similarity in the inclination is due to the declination (DEC) of
SgrA⋆ of -29∘ and a face-on orbit is given by a inclination
$\sim\mathrm{DEC}\pm 90^{\circ}$. The difference in the eccentricity, $e$, and
semi-major axis, $a$, can be explained by the different models used for SgrA⋆.
Palumbo et al. (2019) uses a Schwarzschild black hole (a=0) at an inclination
of 10∘. In this work we use four different physical models: GR black holes
(a=0.6 and a=0.94), a Dilaton black hole and a boson star all at an
inclination of $60^{\circ}$. Due to the larger relativistic effects at higher
inclination (Doppler boosting) the GRRT images show a large left-right
asymmetry e.g., the flux distribution is more compact as compared to black
holes seen at an inclination of $10^{\circ}$. In addition the boson star used
in our work shows the most compact flux distribution around $10\,\mathrm{\mu
as}$ (see panel i in Fig. 4). Since we require in our optimisation to resolve
and recover the compact structure of the black holes and at the same time the
boson star we need to improve the angular resolution (longer base lines) and
the u-v plane coverage (short baselines). Using one satellite this can be
achieved by an elliptical orbit. In our case the perigee is located at a
height of $\sim 1100\,\mathrm{km}$ and the apogee at a height of $\sim
22300\,\mathrm{km}$. The space-ground tracks of this orbit are presented in
panel f of Fig. 2 and the plot shows clearly the improved u-v coverage (short
baselines with $\sqrt{u^{2}+v^{2}}<9\,G\lambda$ and long base lines
$\sqrt{u^{2}+v^{2}}>9\,G\lambda$).
Using this satellite orbit and the observing time span from 4:00-16:00 UT the
reconstructed images for all black holes and the boson star clearly improve
the quality of the reconstructed images (better image metrics as compared to
the EHT 2017 array) and are able to recover the structure seen in their
infinite resolution GRRT counterparts (see Fig. 4).

After the optimisation of the space-EHT e.g., defining the orbit of the space
antenna and the observing time we create synthetic data taking the source
variability during the course of the EHT observation into account. Therefore
we create for each of the space-times under investigation a movie from the
GRRT images which covers 12 h of observations (first 216 GRRT images are used,
see indicated time 12 h time span in Fig. 1). From this movie we create the
synthetic visibilities using the same parameters as used during the orbit
optimisation (see Table 2 and 4). As a consequence of creating the synthetic
data from a variable source the visibility amplitude (VA) of the zero
baselines (ALMA-APEX, JCMT-SMA) is variable and not stationary as in the case
of using an 12 h averaged GRRT image for the data creation. This behaviour is
also true for non-zero baselines and for the closure phases (CP). In Fig. 3 we
compare the VA and CP created from GRRT movie and from its static average
frame for the ALMA-SMT baseline and for the ALMA-LMT-SMT triangle. The
differences in the behaviour of the VA and CP are clearly visible and the most
striking difference can be seen at t=10 UT.

Figure 3: Comparison between synthetic data generation from a dynamical movie
(blue lines) and from its static average frame (black lines) for the
visibility amplitude (top) and closure phase (bottom) for Kerr black hole with
${\rm a}_{*}$=0.6.

##  4 Results

Given the underlying variability in the flux density and also in source size
(see Fig. 1) of the different models, standard VLBI imaging approaches which
assume a static source during the course of the observations are likely to
fail reconstructing an image. Therefore we follow the approach of Johnson et
al. (2017) and use dynamical imaging to reconstruct an average image from the
time variable data. Since we are mainly interested in the average image
obtained during the observation and our GRRT images do not show significant
flaring events666we define a flare as doubling of the flux density from one
frame to another we use the $\mathcal{R}_{\Delta I}$ regularizer with $D_{2}$
distance metric and complex visibilities as data product (see Johnson et al.,
2017, for more details). We reconstructed 24 frames with a duration of 30
minutes each for a total duration of 12 h. For the first initialisation of the
imaging we applied a circular gaussian with FWHM=70 $\mu$as prior image and
for repeated re-initialisation of the imaging we use as prior the previously
obtained average image computed from all 24 frames convolved with the half of
the nominal array resolution. For the dynamical reconstruction we applied 5
re-initialisation loops.

###  4.1 Image Plane comparison

In order to quantify the ability to test different space-times we compute the
structural dissimilarity measure (DSSIM) (Wang et al., 2004) between the GRRT
and reconstructed images. Before computing the DSSIM we perform an image
alignment using the normalised cross correlation coefficient (nCCC) (see,
e.g., Mizuno et al., 2018). Both image metrics are reported in the panels of
Fig. 4 and in Table LABEL:results. In Table LABEL:results we list the DSSIM
and nCCC values computed for the different image combinations and for the
different array configurations. If we could clearly distinguish among
different theories based on the reconstructed images, the smallest DSSIM and
largest nCCC values should be obtained for equal image pairs e.g., boson star
– boson star (the diagonal in Table LABEL:results). However, for the EHT 2017
configuration this is only true for the boson star. Thus based on
reconstructed images obtained from synthetic data which take the source
variability into account it is difficult for the EHT 2017 configuration to
distinguish the spin of Kerr black holes and to differentiate a Kerr black
from a dilaton black hole (see very similar values in

Figure 4: Synthetic black hole images for Sgr A∗ for a Kerr black hole with
${\rm a}_{*}$=0.94 (panels a-c), a Kerr black hole with ${\rm a}_{*}$=0.6
(panels d-f), a non-rotating dilaton black hole (panels g-i) and for a stable,
non-rotating boson star (panels j-l). From left to right and for all rows:
GRRT image (panels a, d, g, j), dynamically reconstructed image with
interstellar scattering (including both, diffractive and refractive scattering
during the generation of the synthetic visibilities) using the EHT 2017
configuration convolved with 75% (red shading) of the nominal beam size (light
grey shading, panels b, e, h, k) and dynamically reconstructed image with
interstellar scattering (including both, diffractive and refractive scattering
during the generation of the synthetic visibilities) using the space-EHT
configuration convolved with 75% (red shading) of the nominal beam size (light
grey shading, panels c, f, i, l)

Table LABEL:results). Including a space antenna improves the ability to
distinguish among the different spacetimes, especially for the Kerr black hole
with high spin. Given the obtained image metrics the space-EHT concept can
distinguish the spin of the Kerr black holes, but it is still difficult to
discriminate between a Kerr black hole with spin ${\rm a}_{*}$=0.6 and a
dilaton black hole (see Table LABEL:results).

By comparing the image metrics obtained for EHT 2017 and the space-EHT concept
two different behaviours can be found: improved image metrics (decreased DSSIM
and increased nCCC) and worsen ones (increased DSSIM and decreased nCCC). For
equal image pairs e.g., dilaton-dilaton, the image metrics for the space-EHT
concept improved as compared to the EHT 2017 (as discussed above). However for
unequal image pairs there are two different behaviours. For example in the
boson star - dilaton case the image metrics improved. In contrast to the
dilaton - boson star pair where the image metrics become worse. This behaviour
could be understood in the following way: The boson star image is very compact
as compared to the dilaton model (see left column in Fig. 4). Due to the
limited resolution of the EHT 2017 array the intrinsically large structure of
the dilaton model is smeared and scattered out to an even larger structure
(compare panel i and j in Fig. 4). However the improved imaging capabilities
of the space-EHT concept leads to a more compact and less smeared out source
structure which is reflected by a better matching between the boson star and
the dilaton black hole. The contrary happens in the dilaton - boson star case:
the EHT 2017 blurs the true boson star structure to a large size which matches
better the dilaton structure as compared to sharper more compact boson star
image provided by the space-EHT concept. As a result the image metrics for
this image pair will become worse as compared to the EHT 2017. A similar
behaviour can be seen for Kerr ${\rm a}_{*}$=0.6 - boson star and to some
extent for dilaton - Kerr ${\rm a}_{*}$=0.94 and Kerr ${\rm a}_{*}$=0.6 - Kerr
${\rm a}_{*}$=0.94. In Appendix B we provide a more detailed study on the
variation of the image metrics with respect to intrinsic source size and array
resolution.

Table 6: Results of the cross-comparison of the synthetic images using the EHT 2017 and the advanced EHT configuration. The values correspond to the DSSIM and the numbers in brackets indicated the normalised cross correlation coefficient (nCCC). Small values for the DSSIM and numbers close to 1 for the nCCC indicate well-matched images (lowest DSSIM values are indicated in bold). | boson star | dilaton | Kerr (${\rm a}_{*}$=0.6) | Kerr (${\rm a}_{*}$=0.94)
---|---|---|---|---
_EHT 2017 configuration_
boson star | 0.03 (0.98) | 0.17 (0.79) | 0.18 (0.81) | 0.10 (0.88)
dilaton | 0.16 (0.83) | 0.13 (0.92) | 0.13 (0.92) | 0.12 (0.93)
Kerr (${\rm a}_{*}$=0.6) | 0.16 (0.85) | 0.14 (0.91) | 0.13 (0.92) | 0.12 (0.94)
Kerr (${\rm a}_{*}$=0.94) | 0.09 (0.89) | 0.18 (0.83) | 0.18 (0.85) | 0.09 (0.93)
_EHT including space antenna_
boson star | 0.03 (0.98) | 0.13 (0.88) | 0.14 (0.85) | 0.09 (0.89)
dilaton | 0.17 (0.78) | 0.10 (0.96) | 0.11 (0.96) | 0.12 (0.88)
Kerr (${\rm a}_{*}$=0.6) | 0.17 (0.80) | 0.11 (0.96) | 0.10 (0.96) | 0.12 (0.91)
Kerr (${\rm a}_{*}$=0.94) | 0.09 (0.89) | 0.13 (0.89) | 0.13 (0.92) | 0.07 (0.96)

###  4.2 Fourier Plane comparison

Given that an interferometer measures the Fourier transform of the brightness
distribution of an astronomical source, a more direct comparison between
images of different space-times can be obtained in Fourier space. The
turbulent nature of the accretion process in the GRMHD simulations manifests
itself in large variations in the total flux density and in its flux density
distribution (see Fig. 1 for Kerr ${\rm a}_{*}$=0.6). In this work we include
the variability of the source in the scoring procedure. Therefore we modify
the scheme used in Event Horizon Telescope Collaboration et al. (2019c) for
the analysis of the recent EHT M87 observations. The main modification is that
the for the comparison with the synthetic data we use a GRRT movie generated
from a series of GRRT images spanning the observing time given by the
synthetic data set. In this work typically 216 frames (for an EHT observation
of 12 h) are used and we slide along our GRRT data series for the different
spacetimes where an increment of 5 frames is used. The increment of 5 frames
is justified by the fact that the typical correlation times of the GRRT images
are around 50 M. This implies that around this time the GRRT images can be
regarded as independent realisations of the accretion flow and thus the
different movies created from the sliding window can be considered as
uncorrelated. The first step in the scoring procedure is to create complex
visibilities from the GRRT movie, taking into account the array configuration,
the observing schedule and interstellar scattering, including both,
diffractive and refractive scattering (see also Table 2 and 4). From the
complex visibilities we compute the visibility amplitude, VA, and from closed
antenna triangles the closure phase, CP. The latter is of great importance in
measuring the structural variation within the source. In the second step we
minimise the $\chi^{2}$ for VA and CP between the snapshots and the individual
frames (images) of the GRRT movie by allowing the total flux, the position
angle and the black hole mass777Actually, we vary the plate scale,
$\mu=m_{bh}/d_{bh}$, during the scoring. Assuming a fixed distance $d_{bh}$ to
SgrA∗, we compute the black hole mass, $m_{bh}$, from $\mu$. to vary. Notice
that once the values for the flux scaling, position angle and black hole mass
are set, they are kept constant for all frames of the GRRT movie in order to
ensure the consistency of the movie. In addition to the image scaling, we
perform antenna gain calibrations. We limit the variation in the individual
antenna gains to be between 50% and 150% in order to avoid the compensation of
structural differences among the models by large gain variations. The scoring
is carried out with the well tested GENA pipeline developed for the image
matching of EHT and VLBA observations (Event Horizon Telescope Collaboration
et al., 2019d, c; Fromm et al., 2019) and we focus our analysis on the
synthetic observations including a space-antenna. The scheme for this kind of
scoring which includes the source variability (hereafter movie scoring) is
illustrated in Fig. 5.

Figure 5: Illustration of the movie scoring scheme. From the GRRT data we
select via a sliding window a set of images from which a movie is created.
From this movie complex visibilities are generated and the $\chi^{2}$ between
the synthetic data is computed while allowing the movie to rotate and vary in
source size and flux density. The $\chi^{2}$ are minimised either using an
evolutionary algorithm (EA) or an MCMC scheme. After the optimisation the
sliding window is advanced until the end of GRRT data set is reached.

In Fig. 6 we show an example for movie scoring for Kerr ${\rm a}_{*}$=0.94
synthetic data to the Kerr ${\rm a}_{*}$=0.94 model. The visibility amplitude
is plotted in the top panel and the closure phase in the second panel. The
blue points correspond to the synthetic data and green ones to the movie. In
the third panel we present the calibrated gains for the antennas involved in
the observations. The bottom panel shows the static average frame from the
Kerr movie (GRRT image left and convolved on the right). The obtained values
for the mass, the flux scaling and position angle are indicated in the top
panel. The recovered values are in good agreement with the injected values. A
more detailed self-test of the movie scoring scheme is presented in the
Appendix C.

Figure 6: Scoring result of the Kerr ${\rm a}_{*}$=0.94 synthetic data to the
Kerr ${\rm a}_{*}$=0.94 model. The panels show from top to bottom the
visibility amplitude, the closure phase, the calibrated gains and the average
image of the Kerr ${\rm a}_{*}$=0.94 movie (GRRT left and convolved right).

Figure 7: Results for the Kerr ${\rm a}_{*}$=0.6 test. The panels show the
distribution of total $\chi^{2}=(\chi^{2}_{VA}+\chi^{2}_{CP})/2$ (top) and the
position angle, $\phi$ (bottom). In each violin the left hand side corresponds
to ${K^{\rm syn}_{0.6}}-K_{0.6}$ and the numbers above the violins indicate
the results of the two-sided K-S test (see text for further details). The red
line in the bottom panel corresponds to the initial position angle, $\phi=0$
of the GRRT images used to create synthetic data for ${K^{\rm syn}_{0.6}}$.

Figure 8: As in Fig. 7, now for the Kerr ${\rm a}_{*}$=0.94 test.

Figure 9: Results for the Kerr ${\rm a}_{*}$=0.6 -dilaton black hole test with
varying black hole mass. The panels show the distribution of total
$\chi^{2}=(\chi^{2}_{VA}+\chi^{2}_{CP})/2$ (top left), the position angle,
$\phi$ (top right) and the black hole mass (bottom). In each violin the left
hand side corresponds to ${K^{\rm syn}_{0.6}}-K_{0.6}$ and the right hand side
to ${K^{\rm syn}_{0.6}}-D_{0}$. The numbers above the violins indicate the
results of the two-sided K-S test. The red line in the middle and right panel
corresponds to the initial position angle $\phi=0$ and initial black hole mass
$M_{\rm bh}=4.14\times 10^{6}M_{\astrosun}$ used for the GRRT images.

During our analysis we fit the synthetic data generated from the first 12 h of
our GRRT data set to the entire data by advancing the sliding window with an
increment of 5 frames. This allows us to statistically quantify the difference
between the synthetic data generated from the first 12 h to the entire GRRT
data set including the source variability. In the next step we keep the
synthetic data and score it against the GRRT images of remaining three
different spacetimes. Finally, we obtain the $\chi^{2}$-distribution for VA,
CP, the position angle (measured from north through east) and the mass
distribution for various combinations of synthetic data and the entire GRRT
image sets for different spacetimes. Based on these distributions we perform a
Kolmogorov–Smirnov (K-S) test to determine if the synthetic data under
investigation is in agreement of being drawn from one of the four spacetimes.
Below we provide a short example which explains the generation of the violin
plots and the obtained p-value from the K-S test. Assume two normal
distributions one with mean at 5.5 and standard deviation of 0.25 and another
one with mean at 5.7 and standard deviation of 0.35. Now draw 20 random
numbers for the the distribution and create a violin plot (the left half of
the violin is the histogram for the first distributions and the right half
corresponds to the second distribution). In this case the violin looks nearly
symmetrical with only a slight shift and the computed K-S test provides a
p-value of 0.49. This value implies that we cannot reject the hypothesis that
the values are from the same “overall” distribution or in your case the
underlying model (spacetime) is the same. (see left most violin in the top
panel of Fig. 7). On the contrary if the mean of the second distribution is
located at 12 the two wings of the violin are disjoint and the K-S test
provides a p-value of 5$\times 10^{-10}$. This value indicates that the two
distributions are not draw from the same “overall” distributions. In our case
this would imply that the underlying model (space-time) is not the same (see
right most violin in Fig. 7).

In the following we label the different spacetimes by their first letter and a
subscript indicates their spin, for example $K_{0.6}$ for a Kerr black hole
with spin ${\rm a}_{*}$=0.6. We indicated the synthetic data by a superscript
“syn” for example ${K^{\rm syn}_{0.6}}$ for the synthetic data of the Kerr
black hole with spin ${\rm a}_{*}$=0.6 generated from the first 12 h. Using
this notation, scoring the synthetic data Kerr ${\rm a}_{*}$=0.6 against the
entire data set of Kerr ${\rm a}_{*}$=0.94 is given by ${K^{\rm
syn}_{0.6}}-K_{0.94}$.

####  4.2.1 Application to Kerr ${\rm a}_{*}$=0.6 (test 1)

In a first application of the movie scoring technique we want to test whether
we can distinguish the synthetic data for a Kerr black hole with spin ${\rm
a}_{*}$=0.6 from a Kerr black hole with spin a∗=0.94, a dilaton black hole and
from a boson star. Therefore, we score the synthetic data for Kerr black with
spina∗=0.6 against the data set for the different space times and perform the
KS test. Given the known black hole mass for the black hole in SgrA⋆ and its
small uncertainty from the GRAVITY experiment (Gravity Collaboration et al.,
2019) we keep for the black hole mass fixed for this first test and perform
the K-S test only on the total $\chi^{2}=(\chi^{2}_{VA}+\chi^{2}_{CP})/2$ and
the position angle.

Figure 7 shows the results for the Kerr ${\rm a}_{*}$=0.6 test. In the panels
the left half of each violin (blue) corresponds to ${K^{\rm
syn}_{0.6}}-K_{0.6}$ and the right half (from left to right) to ${K^{\rm
syn}_{0.6}}-D_{0}$, ${K^{\rm syn}_{0.6}}-K_{0.94}$ and ${K^{\rm
syn}_{0.6}}-B_{0}$, respectively. The number above the violins indicates the
results of the two-sided K-S test. Within each half-violin the long-dashed
lines indicate the mean and the short dashed line denotes the inter-quartile
range. The top panel displays the $\chi^{2}$ distribution, the bottom panel
the distribution of the position angle, $\phi$. The red line in the bottom
panel indicates initial position angle of images which are used to create the
synthetic data set. For ${K_{0.6}}-D_{0}$ the $\chi^{2}$ show a very similar,
only slightly shifted shape. In contrast to ${K_{0.6}}-K_{0.94}$ and
${K_{0.6}}-B_{0}$ where the distributions are only marginally overlapping. The
distributions for the position angle $\phi$ are in all cases overlapping with
the distribution of the truth model (${K^{\rm syn}_{0.6}}-K_{0.6}$, blue
violins).

This behaviour can be explained by the very similar source size for the Kerr
${\rm a}_{*}=0.6$ and dilaton black hole in contrast to Kerr black hole with
spin ${\rm a}_{*}=0.94$ and the boson star (see left column in Fig 4). In
order to improve the $\chi^{2}$ during the optimisation (or MCMC) step the
image is rotated a bit more in the case of the Kerr ${\rm a}_{*}=0.94$ and the
boson star data set. The results of the K-S test can be found in Table
LABEL:ksresults.
The results of the movie scoring indicate that the space-EHT concept can
clearly distinguish fast from slow spinning black holes as well as from boson
stars.

####  4.2.2 Application to Kerr ${\rm a}_{*}$=0.94 (test 2)

The second test we use the synthetic data from the Kerr black hole with spin
${\rm a}_{*}$=0.94 and we test if it can be distinguished from either a Kerr
black hole with spin ${\rm a}_{*}$=0.6, a dilaton black hole or a boson star.
The following movie scorings are computed: ${K^{\rm syn}_{0.94}}-K_{0.94}$
(synthetic data and GRRT data set are from the same spacetime), ${K^{\rm
syn}_{0.94}}-D_{0}$, ${K^{\rm syn}_{0.94}}-K_{0.6}$ and ${K^{\rm
syn}_{0.94}}-B_{0}$. The $\chi^{2}$ and position angle distribution of all
test pairs of models are clearly shifted (see Fig. 8). This behaviour is also
reflected in the small numbers for the K-S test (see Table
LABEL:ksresults).The shift in the $\chi^{2}$ distributions can be explained by
the difference in source size for the Kerr ${\rm a}_{*}$=0.6 and dilaton black
hole. In order to match the source size of the Kerr black hole with spin ${\rm
a}_{*}$=0.94 the dilaton and Kerr black hole with spin ${\rm a}_{*}$=0.6 would
require a smaller black hole mass $m_{\rm bh}<4.14\times
10^{6}\,M_{\astrosun}$. Since we do not allow the black hole mass to adjust
the only way to improve the $\chi^{2}$ is to rotate the GRRT movies. An
improved $\chi^{2}$ is obtained by rotating the GRRT movies were the mean of
the $\phi$ distributions is located at -15∘ ( Kerr black hole with spin ${\rm
a}_{*}$=0.6) and -25∘ (dilaton black hole). Similar, the boson star would
require a larger black hole mass to fit the Kerr ${\rm a}_{*}$=0.94 synthetic
data.
Given the distributions and the result of the K-S test the space-EHT concept
can distinguish among all three models if the truth model is a Kerr black hole
with spin ${\rm a}_{*}$=0.94.

####  4.2.3 Application to Kerr ${\rm a}_{*}$=0.6 with varying black hole mass
(test 3)

In order to test the variations in the black hole mass required to make the
Kerr ${\rm a}_{*}$=0.6 indistinguishable for the space-EHT concept from the
dilaton black hole we performed a third test where we restrict ourselves to
Kerr ${\rm a}_{*}$=0.6 and dilaton black holes and allow the black hole mass
to adjust during the optimisation. The result for this test is presented in
Fig. 9. As expected the adjusted black hole mass for the dilaton black hole
improved the p-values for the K-S test on all quantities ($\chi^{2}_{\rm
tot}$, $\phi$ and $m_{\rm bh}$). However this would require a black hole mass
of $m_{\rm bh}=4.05\times 10^{6}\,M_{\astrosun}$ which is not in agreement
with the current measurements of GRAVITY (Gravity Collaboration et al., 2019).

Table 7: Results of the K-S tests for Kerr ${\rm a}_{*}$=0.6 and Kerr ${\rm
a}_{*}$=0.9 as truth images. The K-S tests are performed on the distribution
of the total $\chi^{2}_{\rm VA+CP}$ (visibility amplitude and closure phases)
and the distribution of the black hole mass, $m_{\rm bh}$. truth model
${K^{\rm syn}_{0.6}}$ (test 1)
---
| ${K^{\rm syn}_{0.6}}-D_{0}$ | ${K^{\rm syn}_{0.6}}-K_{0.94}$ | ${K^{\rm syn}_{0.6}}-B_{0}$
p-value $\left(\chi^{2}\right)$ | $0.46$ | $1.6\times 10^{-9}$ | $1.3\times 10^{-9}$
p-value $\left(\phi\right)$ | $0.55$ | $0.22$ | $0.12$
truth model ${K^{\rm syn}_{0.94}}$ (test 2)
| ${K^{\rm syn}_{0.94}}-D_{0}$ | ${K^{\rm syn}_{0.94}}-K_{0.6}$ | ${K^{\rm syn}_{0.94}}-B_{0}$
p-value $\left(\chi^{2}\right)$ | $7.0\times 10^{-7}$ | $1.0\times 10^{-7}$ | $1.6\times 10^{-9}$
p-value $\left(\phi\right)$ | $1.2\times 10^{-10}$ | $7.5\times 10^{-10}$ | $2.8\times 10^{-4}$
truth model ${K^{\rm syn}_{0.6}}$ (test 3)
| ${K^{\rm syn}_{0.6}}-D_{0}$
p-value $\left(\chi^{2}\right)$ | 0.74
p-value $\left(\phi\right)$ | 0.78
p-value $\left(m_{\rm bh}\right)$ | 0.94

##  5 Discussion and Summary

In this exploratory paper we address the question if an orbiting space antenna
will improve the capabilities of the EHT to distinguish among different space
times around SgrA⋆. Our proposed optimisation procedure suggested an
elliptical orbit with an eccentricity of e=0.5 and a semi-major axis of
a=14900 km. By including the suggested space antenna the baselines of the EHT
array could be extended beyond 10,000 km and thus increase the angular
resolution and the imaging capabilities of the array considerably (see right
column in Fig. 4). As can be seen in Table LABEL:results the space-EHT is able
to distinguish the spin of Kerr black holes (best image metrics can be found
for equal image pairs). For example the DSSIM for the Kerr ${\rm a}_{*}$=0.6 –
Kerr ${\rm a}_{*}$=0.6 dropped from 0.13 to 0.10 while the Kerr ${\rm
a}_{*}$=0.6 – Kerr ${\rm a}_{*}$=0.94 is 0.12. The improvement on the imaging
capabilities of the space-EHT is also noticeable in the change of the computed
nCCC: for Kerr ${\rm a}_{*}$=0.6 from 0.91 to 0.96 while for the Kerr ${\rm
a}_{*}$=0.94 case the nCCC decreased from 0.94 to 0.91. A similar behaviour is
found for the other equal image pairs e.g., Kerr ${\rm a}_{*}$=0.94 – Kerr
${\rm a}_{*}$=0.94 as compared to the non-equal image pairs e.g., Kerr ${\rm
a}_{*}$=0.94 – boson star (see last row in Table LABEL:results). For the ${\rm
a}_{*}$=0.6 Kerr black hole and the dilaton black hole the image metrics
improved on very similar: the DSSIM value: from 0.13 to 0.10 and the nCCC from
0.92 to 0.96. Given these very similar number is currently not possible to
distinguish both spacetimes based solely on their reconstructed images.

To circumvent the limitations of reconstructed average images in
distinguishing different theories of gravity, we additionally performed a
detailed comparison in Fourier space including the source variability.

We created synthetic data from the first 12 h of the GRRT images of the two
different Kerr black holes and scored them against 12 h movies creates from
the GRRT images of all four spacetimes. Fitting the synthetic data for the
Kerr black hole with spin ${\rm a}_{*}$=0.6 against its entire GRRT data set
leads to a $\chi^{2}$ distribution with a mean value of 5.6 and standard
deviation of 0.5 (see blue violins in Fig 7). The position angle distribution
peaks at a value of $\phi=0.7^{\circ}\pm 4^{\circ}$. This value is close the
initial value we used for the creation of the synthetic data and can be
regarded as confirmation of the fitting routine.

We performed two-sided K-S tests to investigate the hypothesis that the
synthetic data is drawn from a spacetime other than Kerr. The result for the
Kerr ${\rm a}_{*}$=0.6 tests reveal very small numbers for the synthetic data
generated from Kerr ${\rm a}_{*}$=0.9 and a boson star, both for the
$\chi^{2}$ and for the position angle (see Fig 7 and Table LABEL:ksresults).
Thus we could reject the null-hypothesis and conclude that Kerr ${\rm
a}_{*}$=0.9 and a boson star can be distinguished from an ${\rm a}_{*}$=0.6
Kerr black hole. The K-S test between the ${\rm a}_{*}$=0.6 Kerr and the
dilaton black holes provides values significantly larger than for the Kerr
${\rm a}_{*}$=0.94 and a Boson star. The mean values of the dilaton
distributions are shifted by roughly one standard deviation compared to the
truth distribution (Kerr ${\rm a}_{*}$=0.6). Given the obtained values, it is
likely (54% level) that we can distinguish an ${\rm a}_{*}$=0.6 Kerr black
hole from a dilaton black hole. In an additional test we allowed the black
hole mass to adjust and in order to probe the mass variation which would lead
to undistinguishable data sets e.g., we can not differentiate between a Kerr
black hole with spin ${\rm a}_{*}$=0.6 and a dilaton black hole. The shift
between the $\chi^{2}$ and $\phi$ distribution decreased which leads to larger
p-values for the K-S test (see Table LABEL:ksresults and Fig. 9). In addition
to the previous test we also obtain the distribution for the black hole mass.
The obtained mean black hole mass for Kerr black hole with spin ${\rm
a}_{*}$=0.6 is $m_{\rm bh}=4.1^{+0.05}_{-0.15}\times 10^{6}\,M_{\astrosun}$
which is in good agreement with used black hole mass during the radiative
transport calculations and a confirmation of the movie scoring method. In
order to make both theories of gravity undistinguishable to 74% the mean black
hole mass should decrease to $m_{\rm bh}=4.02\times 10^{6}\,M_{\astrosun}$.
However, given the mass boundaries from the GRAVITY experiment this black hole
mass is outside the allowed range. Similar to the ${\rm a}_{*}$=0.6 Kerr case,
we perform K-S tests on the fast spinning Kerr black hole with ${\rm
a}_{*}$=0.94. Again, we obtain the distribution for the $\chi^{2}$ and the
position angle $\phi$ and computed the K-S test. All distributions for this
test are clearly offset from the truth distribution (see top panel in Fig. 8),
which is also reflected in the very small numbers obtained for the p-value
(see Table LABEL:ksresults). However, given the small p-values we conclude
that the fast-spinning Kerr black hole can be clearly distinguished from the
slower-spinning Kerr black hole, the dilaton and the boson star.

In summary, we have presented a future EHT concept which includes a space
antenna and tested the capabilities of this new array to investigate the
different possible spacetimes around Sgr A∗ via radio images and complex
visibilities. The orbit of the satellite is computed from a non-linear
optimisation using a genetic algorithm and constraints on the u-v-plane
filling, observation time and improved image metrics computed between the GRRT
image and the reconstructed image. We generated synthetic data for three
different theories of gravity, namely: a Kerr black hole (general relativity),
a dilaton black hole and a boson star. The synthetic data generated was
created from 12 h of GRRT movie in order to properly include the source
variabily and a dynamical image reconstruction was applied following Johnson
et al. (2017) using ehtim. From the dynamical reconstructed images an average
image was computed and compared to the average frame of the GRRT movies using
DSSIM and nCCC metrics. The image plane comparison was accompanied by a more
detailed and robust analysis in the Fourier plane using the newly developed
movie scoring method in GENA. A K-S test was performed on the $\chi^{2}$
position angle and mass distributions, in order to investigate the possibility
of distinguishing among different spacetimes and therefore different theories
of gravity. The space-EHT concept presented in this study has been shown to
both improve the imaging capabilities while including the source variability
of the array and improve our ability to distinguish among (and potentially
exclude) certain solutions and theories of gravity.
Our ability to image time-variable sources and probe the underlying spacetime
will be further improved by the addition of additional ground antennas for
example the African Millimetre Telescope (ATM) (Backes et al., 2016) or
several telescopes placed at dedicated locations across the globe as planed by
the next generation EHT (ngEHT) (Blackburn et al., 2019). Extending the GRRT
images series will allow us to address the source variability on larger non-
overlapping time windows improving the statistics. Increasing the observing
frequency ($\nu>500$ GHz) will reduce the interstellar scattering and also
allow us to study more deeply general relativistic effects on horizon scales
since the obtained image is less effected by the properties of the accretion
flow and the radiation microphysics. However, due to the opacity of the Earths
atmosphere such a concept would require an entirely space-based VLBI concept
(see e.g., Roelofs et al., 2019; van der Gucht et al., 2019).
In addition to horizon scale images of the EHT and future space-EHT concepts
further constraints on the properties of the spacetime around Sgr A⋆ can be
obtained from a pulsar in a tight orbit (orbital period $\sim 1$ yr ) around
the galactic centre (Liu et al., 2012). Combined, both measurements e.g.,
horizon scale images and pulsar timing would allow tightly constraints of the
properties of spacetime around SgrA⋆ (Psaltis et al., 2016).

###### Acknowledgements.

CMF thanks E. Ros, R. Porcas and D. Palumbo for fruitful discussion and
comments on the manuscript. Support comes from the ERC Synergy Grant
“BlackHoleCam - Imaging the Event Horizon of Black Holes” (Grant 610058). CMF
is supported by the black hole Initiative at Harvard University, which is
supported by a grant from the John Templeton Foundation. ZY acknowledges
support from a Leverhulme Trust Early Career Fellowship.

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##  Appendix A Influence of the orbital parameters on the imaging capabilities

In Fig. 10 we illustrate the impact of the orbital parameters on the imaging
capabilities of the space-EHT concept and the tradeoffs between u-v coverage
and array resolution. We keep the orientation parameters of the orbit fixed to
the results of our orbit optimisation (see Table 5) and change the semi-major
axis, $a$, and eccentricity, $e$.

Figure 10: Reconstruction of a boson star image for SgrA⋆ (ground truth image
in panel a) for three different satellite orbits; small orbit (second column),
intermediate orbit (third column) and large orbit (fourth column). The columns
show from top to bottom the reconstructed image convolved with 75% of the
nominal array resolution, the uv-coverage (white: ground array baselines, red:
space baselines) and the satellite orbit as seen from SgrA⋆.

An increase in the semi-major axis, $a$ (see panel j in Fig. 10), leads to
higher resolution (see beam in panel d in Fig. 10). At the same time the
larger semi-major axis, $a$, which corresponds to a larger orbital period
$T_{p}=2\pi\sqrt{a^{3}/\left(GM_{\rm Earth}\right)}$ reduces the u-v coverage
(see panel g in in Fig. 10) . This leads to a worse image metric as for
example an medium size orbit (compare image metrics in panel c and d in Fig.
10). On the other hand, a small semi-major axis (see panel h in Fig. 10) leads
to a dense u-v coverage (panel e in Fig. 10) but does not provide high angular
resolution (see panel b in Fig. 10). A trade-off between resolution and u-v
coverage is an intermediate size semi-major axis (see third column in in Fig.
10). For each of the three satellite orbits the image metrics are calculated
which prioritise the orbit with the best image metrics here the orbit with a
semi-major axis of $a=14861$ km and an eccentricity, $e=0.5$.

##  Appendix B DSSIM and nCCC

In order to explore the variation of the DSSIM and the nCCC we performe a
small parameter space study using averaged GRRT images. The GRRT images are
taken from Kerr BHs with spin ${\rm a}_{*}$=0.6 and ${\rm a}_{*}$=0.94 as well
as from the dilaton black hole. To mimic different observing arrays with
improved imaging capabilities we convolve the average images with different
beam sizes888we applied a circular gaussian ranging from 30 $\mu$as to 1
$\mu$as. The convolved images are compared to the ground truth GRRT images and
the DSSIM and the nCCC values are computed.

Figure 11: DSSIM and nCCC variation study. Top panel: ground truth images for
Kerr ${\rm a}_{*}$=0.6, Kerr ${\rm a}_{*}$=0.94 and dilaton black holes (left
most column) and convolved with decreasing beams (30 $\mu$as to 1 $\mu$as).
Middle panel: DSSIM and nCCC calculations using the average image of Kerr BH
with ${\rm a}_{*}$=0.6 as ground truth image. Bottom panel: DSSIM and nCCC
calculations using the average image of Kerr BH with ${\rm a}_{*}$=0.94 as
ground truth image.

In Fig. 11 we show the results for this study. In the top panel we display the
average GRRT images for the Kerr black holes (first and second row) and for
the dilaton black hole (third row). Note that the images are individually
normalised by their flux density maximum. The first column presents the ground
truth GRRT image followed by a series of convolved images using beams with
decreasing size (beam size is indicated above each image). With decreasing
beam size (equal to the improved imaging capabilities of the array) the images
become sharper and finer image features become visible. In the second panel of
Fig. 11 we compute the image metrics assuming that the ground truth image
corresponds to a Kerr BH with spin ${\rm a}_{*}$=0.6. In the panel the red
colour corresponds to the DSSIM and blue to the nCCC while the line style
indicates the image pairs, e.g., red dashed line stands for the DSSIM computed
between Kerr BH with ${\rm a}_{*}$=0.6 and Kerr BH ${\rm a}_{*}$=0.94. As
expected the DSSIM values decrease with smaller beam sizes while the nCCC
increases for all image pairs. However at a beam size of $\sim 16\,\mu$as the
DSSIM values for the Kerr BH with ${\rm a}_{*}$=0.94 increases and at the same
time the nCCC decreases. This behaviour can be understood by the different
source sizes of the ground truth images (see left most column in the top
panel). The compact source structure of the Kerr BH with ${\rm a}_{*}$=0.94 is
smeared out to larger extents during the convolution with beam with size
$>16\,\mu$as which matches better the ground truth structure of the Kerr BH
with ${\rm a}_{*}$=0.6. Thus, the DSSIM is decreasing while the nCCC is
increasing. However at a beam size of $\sim 16\,\mu$as the behaviour is
inverted. For the models with similar ground truth sizes the DSSIM is
monotonically decreasing while the nCCC is continuously increases.
If the size of the ground truth image is smaller than the intrinsic size of
the models to which it is compared, e.g., Kerr BH with ${\rm a}_{*}$=0.94 and
Kerr BH ${\rm a}_{*}$=0.6 no change in the DSSIM and nCCC behaviour is found:
the DSSIM is always decreasing and the nCCC is continuously increasing with
decreasing beam size (see bottom panel in Fig. 11).

##  Appendix C Movie scoring self-test

To valid the developed movie scoring method we perform a self-test. For the
self-test we generate synthetic visibilities from the Kerr black hole with
spin ${\rm a}_{*}$=0.94. The start frame for the synthetic data generation is
shifted by 1.7 hours (30 frames). During the generation of the synthetic data
we take interstellar scattering (including both, diffractive and refractive
scattering), thermal noise and gain variations into account (see also Table
4). For a successful self-test the movie scoring method should recover the
correct starting frame, the black hole mass and the orientation of the images
used during the synthetic data generation. We during the radiative transfer we
use a black hole mass of $4.14\times 10^{6}\,M_{\astrosun}$ and an orientation
angle of $\phi=0^{\circ}$. During the self-test we allow the black hole mass,
$m_{\rm bh}$, the rotation angle, $\phi$ and the flux normalisation, S to
adjust during the scoring. For the optimisation we apply a MCMC method
(Foreman-Mackey et al. 2013; Foreman-Mackey 2016) using 100 MCMC walkers for
500 iterations and perform gain-calibration.

Figure 12: Posterior distributions for the black hole mass, $m_{\rm bh}$, the
position angle, $\phi$, and the flux scaling, $S_{\rm scale}$. The sliding
window is located at $\Delta t=1.7$ h (30 frames) recovering the time range
used for the synthetic data generation (see Text).

The movie scoring method successfully recovered the start frame and in Fig. 12
we present the posterior distribution for the black hole mass, the orientation
angle and the flux scaling for the GRRT movie starting at frame 30. The black
hole mass and the orientation angle are recovered within $1\sigma$ proofing
the applicability of the developed method to recover black hole parameters
from a source with varying flux density and structure. The posterior
distribution of the flux scaling, $S_{\rm scale}$, reflects the influence of
interstellar scattering and antenna gain variations taken into account during
the synthetic data generation. Interstellar scattering smears out the flux
density, thus reducing the measured visibility amplitude as compared to an un-
scattered case. In order to match the visibility amplitude during the scoring
the flux scaling factor is smaller than one. The additional variations due to
the antenna gain variations are compensated by the applied self-calibration.
In addition we show in Fig. 13 the posterior distribution for the scaling
parameters ($m_{\rm bh}$, $\phi$ and $S_{\rm scale}$) obtained from the entire
data set for the Kerr black with ${\rm a}_{*}$=0.94. The posterior
distribution for the black hole mass, $m_{\rm bh}$, and the flux scaling,
$S_{\rm scale}$ show a similar distribution as for the best starting frame
model (see Fig. 12) except for the three times larger uncertainties for the
black hole mass. However the distribution of the position angle, $\phi$, shows
a second local maximum around $\phi=3^{\circ}$. This can be explained by the
variability of the source or more precise the variation of the position angle
of the flux centroid. In Fig. 14 we show the variation of the image centroid
position angle relative to the data set used for the synthetic data. For
sliding windows different that the one used for the data generation (indicated
by the black arrow in Fig. 14) the centroid position angle is lager angle (see
green dashed line in Fig.14). During the optimisation within the movie scoring
method this angle difference is compensated by a rotating the images which
explains the second maximum in the position angle posterior distribution in
Fig. 13.

Figure 13: Posterior distributions for the black hole mass, $m_{\rm bh}$, the
position angle, $\phi$, and the flux scaling, $S_{\rm scale}$ for the entire
Kerr black hole ${\rm a}_{*}$=0.94 data set.

Figure 14: Variation of the image centroid position angle relative to the one
from the sliding window used for the generation of the synthetic data
(indicated by the black arrow). The mean centroid position angle for a second
sliding window with 10 hour offset is indicated in green.