# On the principle of Astrometric Gravitational Wave Antenna

Mariateresa Crosta<EMAIL_ADDRESS>Astrophysical Observatory of
Turin (OATo) - INAF, via Osservatorio 20, I-10025 Pino Torinese (TO), Italy
Mario G. Lattanzi  Astrophysical Observatory of Turin (OATo) - INAF, via
Osservatorio 20, I-10025 Pino Torinese (TO), Italy  Christophe Le Poncin-
Lafitte  SYRTE, Observatoire de Paris, Université PSL, CNRS, Sorbonne
Université, LNE, 61 avenue de l’Observatoire, 75014 Paris, France  Mario Gai
Astrophysical Observatory of Turin (OATo) - INAF, via Osservatorio 20, I-10025
Pino Torinese (TO), Italy  Qi Zhaoxiang  Shanghai Astronomical Observatory,
CAS 80 Nandan Road, Shanghai 200030, China  Alberto Vecchiato  Astrophysical
Observatory of Turin (OATo) - INAF, via Osservatorio 20, I-10025 Pino Torinese
(TO), Italy

###### Abstract

The direct detection of gravitational waves by ground-based optical
interferometers has opened a new window in astronomy. Nevertheless, as these
detectors are a combination of two Michelson-Morley like baselines, their
sensitivity for determining the incident direction of a gravitational wave is
quite weak compared to current high-precision of space astrometry. We
therefore present a novel concept for a gravitational wave antenna in space
that uses close pairs of point-like sources as natural sensors to record and
characterize the very tiny variations in angular separations induced by a
passing gravitational wave, thus operating complementarily to linear arm
detectors and enabling to gain informations on the gravitational wave incoming
direction. Indeed, the proposed astrometric gravitational wave observable
builds on methods of relativistic astrometry that can substantially enhance
the strength of gravitational wave signals by exploiting to the fullest the
telescope optical resolution and, at same time, provide a powerful tool for
identifying the direction to the sources that originated the gravitational
wave by monitoring close pairs of stars. Furthermore, the use of two local-
line-of-sights in a differential formulation avoids issues related to high
order modeling of the local (Solar System) background geometry and the need
for accurate satellite ephemeris and attitude.

##  I Introduction

Experimental confirmation of gravitational waves (GWs) through the LIGO
(Abbott et al., 2016; Aasi et al., 2015) and VIRGO (Acernese et al., 2019)
antennas have gained great impulse to the search and characterization of
possible GW sources. New ground-based experiments are joining (Akutsu et al.,
2019) or in the making (Punturo et al., 2010; Abbott et al., 2017; Kawamura et
al, 2019) and the LISA mission (Amaro-Seoane et al., 2017) will soon implement
similar concepts in space. The prime objective is the complete description of
GWs, i.e., to determine their amplitude and frequency, and then pinpoint their
directions for multi-waveleng “profiling”. A recent review on gravitational-
wave physics and astronomy in the present decade can be found in Bailes et al.
(2021).

Astrometric observations are primarily targeted for the accurate determination
of directions to the incoming photons and their change with time. Therefore
they collect photons that have interacted with different time-dependent
gravitational fields along their propagation. With the advent of highly
accurate astrometric (angular) and radial velocity measurements in space,
observation reduction models compliant with General Relativity (GR) have
become a necessity. This is because, when compared to the targeted measurement
precision levels, the amount of space-time curvature due to the Sun and to all
of the other relevant Solar System (SS) masses (including the Earth-Moon
system) that are affecting incoming photons and space “observers” alike can no
longer be ignored. Weak gravity regime influences electromagnetic propagation
on a much wider domain than its strong counterpart.
A gravitational wave detection via astrometry was explored by several authors
(Braginsky et al., 1990; Pyne et al., 1996; Kaiser & Jaffe, 1997; Gwinn et
al., 1997; Book & Flanagan, 2011; Mihaylov et al., 2018). In practice, the
main issue is to consider deflection extra-shifts induced by passing GWs on
photons propagating to an observer from within the SS. Such a detection is not
considered promising, since it requires, at lower order, the nanoarcsecond
level of accuracy (Damour & Esposito-Farèse, 1998; Crosta et al., 1998;
Schutz, 2010) and depends on the knowledge of the satellite attitude. So far,
preliminary analysis has been focused on periodic GW signals with period
shorter than the Gaia operating time (Moore et al., 2017; Klioner, 2018) or on
secular effects (Mignard & Klioner, 2012) in QSO proper motions for longer
period, i.e. for ultra-low GW frequencies.
As a matter of fact, the Gaia-like astrometric observable is the direction
cosine between incoming stellar light and the observer attitude-tetrad (Crosta
et al., 2017). Its strict application to a passing GW generates serious
shortcomings. The addition of a correction to the direction cosine solely due
to the passing GW signal, ignoring the instrumental contribution of the
satellite uncertain attitude, would be not the optimal strategy for our
purpose, i.e. to find the incoming GW signal, as it would impose a requirement
on the accuracy of the attitude knowledge to implement in-orbit. Besides the
above, disentagling the GW signals requires that the SS background metric
should be consistently developed to account for possible background (natural)
“noises”, thus preventing systematic effects in modelling our astrometric
observable. Hence, since $v/c\sim 10^{-4}$ rad, being $v$ the typical velocity
of each relevant SS metric source, one should retain terms in the null
geodesic solution up to $(v/c)^{4}$ at best, i.e., at the nanoarcsecond level
(nas).
Here we present a novel idea of an $\it{Astrometric~{}GW~{}Antenna}$ that uses
angles, i.e. differential astrometry, instead of linear baselines as the
current “spatial” antennas taking advantage from the relativistic astrometric
techniques successfully developed for Gaia (ESA, Gaia Collaboration, 2016;
Crosta et al., 2017).

To overcome the limitations inherent in the measurement of a single light
direction, in the following we introduce a different fundamental observation
equation for the astrometric GW detection based on the angle relative to pairs
of local line-of-sights. Finally, we also give a first evaluation of its
potential impact on GW science and its practical feasibility, in the light of
new, specialized and much improved concepts for space astrometry missions
(Lattanzi, 2012; Gai et al., 2020a, b).

##  II The fundamental observation equation for the space-born astrometric
gravitational wave antenna

Given the global metric due to both SS sources and passing/standing GW
perturbations at the observer’s location in the form

${g}_{\alpha\beta}={g}^{SS}_{\alpha\beta}+{h}_{\alpha\beta}^{GW}=\eta_{\alpha\beta}+\sum_{(a)}{h}_{(a)\alpha\beta}^{SS}+{h}_{\alpha\beta}^{GW}+O(h^{2})~{},$
(1)

where $\eta_{\alpha\beta}$ is the flat Minkowskian metric, with signature
(-,+, +,+), the cosine of the angle between two observed light directions
${\bar{\ell}}^{\alpha}_{1}$ and ${\bar{\ell}}^{\alpha}_{2}$ writes (see
appendix A for the derivation)

$\cos\psi_{1,2}={g}_{\alpha\beta}({\bar{\ell}}^{\alpha}_{1}{\bar{\ell}}^{\beta}_{2})_{obs}~{},$
(2)

where ${\bar{\ell}}^{\alpha}$ is the null tangent unit four-vector projected
on the rest space of the local barycentric observer, namely, for of our solar
system, the observer at rest relative to the barycentric celestial reference
system (BCRS) with coordinates $(t,x^{i})$ (Crosta, 2019, and references
therein). The differential nature of the measurement greatly relaxes
requirements on a precise satellite attitude and payload (thermal and
mechanical) stability.

Similarly, stellar light directions can be separated into the SS part (due to
the background metric) plus a perturbation shift, i.e. $\delta\ell^{\alpha}$,
attributed purely to the passing GW:

${\bar{\ell}}^{\alpha}_{obs}=\bar{\ell}^{\alpha(SS)}+\delta\ell^{\alpha(GW)}+O(\delta\ell^{2}).$
(3)

Then, the cosine can be further simplified if
$\delta\ell^{(GW)}\sim\epsilon^{4}$ as follows

$\cos\psi_{1,2}=\cos\psi_{1,2}^{SS}+\eta_{\alpha\beta}({\bar{\ell}}^{\alpha}_{1_{0}}\delta\ell_{2}^{\beta(GW)}+{\bar{\ell}}^{\alpha}_{2_{0}}\delta\ell_{1}^{\beta(GW)})_{obs}+h^{GW}_{\alpha\beta}{\bar{\ell}}^{\alpha}_{1_{0}}{\bar{\ell}}^{\beta}_{2_{0}}+O(\epsilon^{5})+O(h^{2}),$
(4)

where ${\bar{\ell}}^{\alpha}_{i_{0}}$ represents the unpertubed light
direction to star i=1,2.

It is straightforward to check that the cosine of the total angle, cannot be
simply split into a background direction cosine plus its GW analogue (see
appendix B for details). On the other hand, since the passing GW produces an
extra shift on the light deflection, we expect a perturbation
$\delta\psi^{GW}$ to the undisturbed angle between two light directions.
Then, we can expand our observation equation as follows:

$\cos(\psi_{1,2}^{SS}+\delta\psi_{1,2}^{GW})=\cos(\psi_{1,2}^{SS})\cos(\delta\psi_{1,2}^{GW})-\sin(\psi_{1,2}^{SS})\sin(\delta\psi_{1,2}^{GW})=\cos\psi_{1,2}^{SS}+F^{GW}_{1,2},$
(5)

where
$F^{GW}_{1,2}\equiv\eta_{\alpha\beta}({\bar{\ell}}^{\alpha}_{1_{0}}\delta\ell_{2}^{\beta(GW)}+{\bar{\ell}}^{\alpha}_{2_{0}}\delta\ell_{1}^{\beta(GW)})_{obs}+h^{GW}_{\alpha\beta}{\bar{\ell}}^{\alpha}_{1_{0}}{\bar{\ell}}^{\beta}_{2_{0}}+O(\epsilon^{5})+O(h^{2})$.

Since $\delta\psi^{GW}\ll 1$, the previous equation reduces to consider the
unknown angle due to the passing GW, i.e.

$\delta\psi_{1,2}^{GW}=-\frac{F^{GW}_{1,2}}{\sin(\psi_{1,2}^{SS})}.$ (6)

The new expression (6) deduced from observation equation (13) depends on the
observed perturbation to the $\psi_{1,2}^{SS}$ angle caused by the passing GW
in the $F^{GW}_{1,2}$ term; the observability of this angular perturbation can
be boosted through the factor (sin$(\psi_{1,2}^{SS}))^{-1}$ that acts as a
“signal amplifier” for the GW detection, being mainly limited by the resolving
power of an astrometric telescope that never goes to zero because of physics.
As we are putting forth a novel operational principle for measuring GWs that
takes great advantage from the quantity $\psi^{SS}_{1,2}$, here it will
suffice to indicate how that can be estimated/derived from actual measurements
of the pair separation. What we measure is actually the angle
$\psi_{1,2}(t_{i})$ between point-like sources ’1’ and ’2’ at time $t_{i}$.
The measurements $\psi_{1,2}(t_{i})$ are taken with a high cadence, i.e., with
frequency $\omega_{S}\gg\omega_{GW}$, the oscillating frequency of the GW,
assumed ’monochromatic’. This last condition ensures that the Nyquist-Shannon
(sampling) theorem is satisfied and, at the same time, sufficient statistics
is built to beat (single) measurement noise.
Let’s then take the average over N ($\gg$ 1), i.e. the number of samples of
the pair angular distance taken over a session, of the measurements
$\psi_{1,2}(t_{i})$. We have: $<\psi_{1,2}(t_{i})>_{N}$
$=<\psi^{SS}_{1,2}>_{N}+<\delta\psi^{GW}_{1,2}(t_{i})>_{N}$
$\simeq\hat{\psi}^{SS}_{1,2}$, as we can make the average
$<\delta\psi^{GW}_{1,2}(t_{i})>_{N}$ as small as needed.

With the evaluation of the unperturbed angular separation of our antenna arm
(the point-like pair) directly from the observations, as shown above, it
follows that:
$\delta\hat{\psi}^{GW}_{1,2}(t_{i})\equiv\psi_{1,2}(t_{i})-\hat{\psi}^{SS}_{1,2}$.
This is the GW perturbation to the antenna angle as estimated directly from
the measurements that can be actually used, together with
$\hat{\psi}^{SS}_{1,2}$, in Eq.(6) to build the observation equations from
which strength and direction to the GW source can be derived. The measurement
and initial data processing strategy/protocol just sketched supports the
theoretical practicability of our concept for an astrometric GW antenna.
Equation (6) requires to define (i) the $h^{GW}$ at the observer, (ii) a pair
of unperturbed local line-of-sights, $\bar{\ell}^{\alpha}_{obs}$, from the
geodesic equation with the metric (1), and (iii) the corresponding shift due
to the passing GW.

The general form of the GW perturbation can be expressed as a function of
argument $\tilde{k}_{\alpha}x^{\alpha}$, namely
$h_{ij}^{GW}(\tilde{k}_{\alpha}x^{\alpha})$ with tangent vector
$\tilde{k}^{\alpha}=\tilde{k}^{0}\partial_{0}^{\alpha}+\tilde{k}^{i}\partial_{i}^{\alpha},$
where $p^{i}=\tilde{k}^{i}/\tilde{k}^{0}$ is the direction of the GW
propagation.

Finally, in the linearized regime, the gravitational wave shift is recovered
within the suitable astrometric models, e.g., those adopted for the data
analysis and processing of the Gaia satellite (Crosta et al., 2017, and
reference therein), via the geodesic integration (see appendix C), namely:

$\int^{\sigma_{*}}_{\sigma_{obs}}d\delta\ell^{i}=\frac{\bar{\ell}^{i}_{{}_{0}}+p^{i}}{2(1+p\cdot\bar{\ell}_{{}_{0}})}\bar{\ell}^{j}_{{}_{0}}\bar{\ell}^{k}_{{}_{0}}\int^{\sigma_{*}}_{\sigma_{obs}}dh_{jk}^{(GW)}(\sigma)-\frac{\bar{\ell}^{j}_{{}_{0}}}{2}\int^{\sigma_{*}}_{\sigma_{obs}}dh_{ij}^{(GW)}(\sigma)+O(\epsilon^{5}),$
(7)

that in the far away zone reduces to

$\delta\ell^{i}=\frac{\bar{\ell}^{i}_{{}_{0}}+p^{i}}{2(1+p\cdot\bar{\ell}_{{}_{0}})}\bar{\ell}^{j}_{{}_{0}}\bar{\ell}^{k}_{{}_{0}}h_{jk(obs)}^{(GW)}-\frac{\bar{\ell}^{j}_{{}_{0}}}{2}h_{ij(obs)}^{(GW)}+O(\epsilon^{5}),$
(8)

explicitly showing that $\delta\ell^{i}\propto h^{GW}$, and in agreement with
what found in Pyne et al. (1996) and Book & Flanagan (2011). In such a case,
our expression (6) then becomes

$\displaystyle\delta\psi_{1,2}^{GW}$ $\displaystyle=$
$\displaystyle-\frac{h^{GW}_{ij(obs)}}{2\sin(\psi_{1,2}^{SS})}\left\\{\frac{[(\bar{\ell}_{1_{0}}\cdot\bar{\ell}_{2_{0}})+(\bar{\ell}_{1_{0}}\cdot
p)]\bar{\ell}^{i}_{2_{0}}\bar{\ell}^{j}_{2_{0}}}{(1+p\cdot\bar{\ell}_{2_{0}})}+\frac{[(\bar{\ell}_{2_{0}}\cdot\bar{\ell}_{1_{0}})+(\bar{\ell}_{2_{0}}\cdot
p)]\bar{\ell}^{i}_{1_{0}}\bar{\ell}^{j}_{1_{0}}}{(1+p\cdot\bar{\ell}_{1_{0}})}\right\\}$
(9) $\displaystyle+O(\epsilon^{5})$

which shows also the dependence on the scalar products between the SS
direction of propagation of photons from stars $1,2$ (i.e., alternatively, the
angle $\psi_{1_{0},2_{0}}$) and that of each star direction with respect to
the GW source (angles $\psi_{1_{0},p}$ or $\psi_{2_{0},p}$).

##  III The operating principle of the GW astrometric antenna

Equation (9) above shows the operating principle of the astrometric antenna,
and clearly demostrates its direct relation with the direction to an incoming
GW and, therefore, the potential to pinpoint its source. For, the GW term
$h_{ij}$ (and its time variation) will mostly characterize the detection (the
amplitude and phase term), while the factor within the curly brackets will
assume specific patterns according to the direction of the incoming GW
relative to the observer (spatial) orientation, i.e. the
${\mathbf{x}},{\mathbf{y}},{\mathbf{z}}$ triad depicted in Fig. 1 (a). In the
“Transverse and Traceless” (TT) standard gauge the GW components are
$h_{0i}=0$, $\delta^{ij}h_{ij}=0$, $\delta^{ij}h_{jk,i}=0$, with only two
independent degrees of freedom, the two amplitudes $A_{+}$ and $A_{\times}$.

Taking advantage of the property that in the TT gauge only the components
perpendicular to the direction of propagation survive, below we show, without
loosing too much to generality, how a GW reaching a space-born observer from
within the SS can be astrometrically measured.
Therefore, to provide quantitative examples suggestive of the potential of Eq.
(9), it will suffice to consider a plane GW, of a given frequency $\omega$,
linearly polarized (i.e., $A_{\times}=0$) , traveling in the direction
$\vec{p}$ assumed aligned with the direction $\bar{\ell}_{1_{0}}$, i.e., the
local (SS) unperturbed direction to star 1. This is the same direction at
which line-of-sight (LOS) ${\mathbf{z}}$, of a three-LOS telescope
materializing the “observer” of Fig. 1(a), is pointing. Also,
$\bar{\ell}_{2_{0}}$, the local direction to star 2, is at a very small
angular separation to $\bar{\ell}_{1_{0}}$, i.e.,
$\cos(\psi_{1_{0},2_{0}})\sim 1$. Similarly to the LOS in the ${\mathbf{z}}$
direction of our schematic 3-LOS (or 3-way) telescope, one can align the other
two viewing directions, ${\mathbf{x}}$ and ${\mathbf{y}}$, along the local
directions $\bar{\ell}_{3_{0}}$ and $\bar{\ell}_{5_{0}}$ to stars 3 and 5,
respectively (Fig. 1 (a)), while directions $\bar{\ell}_{4_{0}}$ and
$\bar{\ell}_{6_{0}}$, to stars 4 and 6, are the corresponding close optical
companions. Close stellar (or stellar-like) pairs here means that we are
always concerned with angles ${\psi}_{i_{0},j_{0}}$ $\lesssim$ 0.01 arcsecond,
a number representative of the operational resolution limit reached with
telescopes already operating (e.g. HST) or that will soon operate in space
(like, e.g., NASA’s NextGen Space Telescope, ESA’s Euclid or CNSA’s CSST, see
further below for more) at optical wavelengths, i.e. $\geq 550$ $nm$,
including the near-IR (to $\sim 2$ micron).

$A_{+}$ | $\psi_{i_{0},j_{0}}$ | $max(\delta\psi^{GW}_{1,2})$ | $max(\delta\psi^{GW}_{3,4})$ | $max(\delta\psi^{GW}_{5,6})$
---|---|---|---|---
(radians) | (”) | ($\mu$as) | ($\mu$as) | ($\mu$as)
$10^{-18}$ | 0.01 | 4.12$\times 10^{-15}$ | 5.12 | 5.12
$10^{-19}$ | 0.01 | 4.12$\times 10^{-16}$ | 0.51 | 0.52
$10^{-20}$ | 0.01 | 4.12$\times 10^{-17}$ | 5.12 $\times 10^{-2}$ | $5.12\times 10^{-2}$
$10^{-21}$ | 0.01 | 0 | $5.12\times 10^{-3}$ | $5.12\times 10^{-3}$
$10^{-18}$ | 0.001 | 4.12 $\times 10^{-16}$ | 51.57 | 51.57
Table 1: Angular perturbations, $max(\delta\psi^{GW}_{i,j})$ ($\mu$as), for
different linear strains of amplitude $A_{+}$ (radians) propagating along
${\mathbf{z}}$. The values derive from Eq. (9) after setting the three antenna
arms $\psi_{i_{0},j_{0}}$ to the values in the second column (in arcsec). As
expected, decreasing $\psi_{i_{0},j_{0}}$ by one order of magnitude, the
corresponding $\delta\psi^{GW}_{i,j}$ is amplified by the same amount. This is
the case for the values reported in the last row compared to its direct
analogue in first. Note that along the GW direction of propagation
$\delta\psi^{GW}_{1,2}$ is not null, as the direction to star 2, forming the
1-2 pair, although quite small, is not coincident with the LOS to star 1.

We are now able to provide actual examples utilizing representative cases of
GW strains known from the literature. The first column of Table 1 presents the
amplitude, $A_{+}$, of metric perturbations from possible GW sources as
described in Thorne (1987) and Colpi & Sesana (2017). The last three columns
provide, for each $A_{+}$, the maximum (angular) perturbations resulting on
the local unperturbed (angular) separations of the pairs (the ”arms” of the
astrometric antenna) along the three telescope axes (Fig. 1 (a)). The
perturbation signals can be calculated from Eq.s (9) for a plane GW traveling
along the ${\mathbf{z}}$ direction and ${\psi}_{i_{0},j_{0}}$ set to the
values in Table I. Gravitational strains as large as $h\sim 10^{-18}$ are
associated with SN core collapse events, thus, they are short lived and at
frequencies $\omega\sim$ 10${}^{3}Hz$. To the range of high frequency sources,
$\omega$=[10 – 1,000] $Hz$, belong also the cases of coalescing compact binary
systems (NS-NS, pairs of stellar black holes, BH∗-BH∗, or NS-BH∗). Besides, at
$\omega$ $<$ 1 $Hz$ and in a range of characteristic strain amplitudes
spanning 3 orders of magnitude, from $\sim$ 10-18 to 10-21, one finds not only
coalescing super-massive BH’s, but also the significantly more numerous
population of resolved and unresolved Milky Way binaries, with at least one
degenerate companion. Figure 1 (b) shows 5 $ms$ (i.e. 5 times the simulated
period) of the angular perturbation $\delta\psi^{GW}_{i,j}$ experienced by the
three ”arms” (the angles ${\psi}_{i_{0},j_{0}}$) under the strain of the high
frequency-high amplitude case mentioned before. As expected, with a GW
propagating in the direction of the ${\mathbf{z}}$-axis, the 1-2 stellar pair
is practically unperturbed, while all the action is with the pairs along the
$\mathbf{x}$ and $\mathbf{y}$ axes.

(a)

(b)

(c)

Figure 1: In (a) a possible configuration of a 3-LOS telescope, the
($\mathbf{x}$,$\mathbf{y}$, $\mathbf{z}$) triad, with respect to the chosen
directions: the local (SS) vectors $\bar{\ell}_{3_{0}}$, $\bar{\ell}_{5_{0}}$
and $\bar{\ell}_{1_{0}}$ to stars 3,5 and 1, respectively. The angles
${\psi}_{i,j}$, representing the instantaneous angular distances of the
stellar pairs are all assumed small at $\sim 0.01^{\prime\prime}$. With the
same colour coding, the other two panels show, fixed the resolution limit, the
astrometric signals $\delta\psi^{GW}_{i,j}$ (from Eq. (9)) caused by a GW
passing along: (b) the $\mathbf{z}$ direction (green line) with $A_{+}\sim
10^{-18}$ and frequency $10^{3}Hz$; (c) the $\mathbf{x}$ direction (red line)
with amplitude $A_{+}\sim 10^{-19}$ and frequency 0.1 $Hz$.The other curves
indicate the GW-induced signals along the other two LOS’s of the astrometric
antenna.

If statistically meaningful measurements of variation of angular separations
of $\sim$ 5 $\mu as$, as in Fig. 1 (b), although possible, are yet to be
proven, variations 10 times larger are already within the capabilities of
present-day, or soon-to-be, space astrometry missions, like those mentioned
earlier (see Lattanzi (2012) and references therein). This is possible also
because actual (angular) resolution limits, governing the amplifications of
the perturbation signal associated with the GW (Eq.(9)), can be made
significantly higher by utilizing optimal calibrations procedures (Bernacca et
al., 1993, 1995; Steffen et al., 2001). Signal frequencies of 1 $kHz$, as in
Fig. 1 (b) are hard to achieve with high signal-to-noise ratio, as sampling
above the Nyquist-Shannon frequency would call for integration times shorter
than 0.5 $ms$. Typical ”fast” exposure times are at 25 $ms$ in the case of the
astrometer FGS aboard HST (Bucciarelli et al., 1994, and references therein),
and only 100 $ms$ on Euclid (Bosco et al., 2018, and references therein),
clearly not enough. And, simply allowing for much shorter exposure times might
not improve things as, given the size of the telescope mirrors on these
missions, it might become difficult to reach sufficiently faint magnitudes,
thus forcing the 3-LOS telescope to limit its orientations to directions with
bright stellar pairs.

Besides, as mentioned earlier, GW sources with similar amplitudes, or even
larger, are expected at more comfortable frequencies, making this scenario a
clear case for GW science with an astrometric antenna. Figure 1(c) illustrates
the progress of $\delta\psi^{GW}_{i,j}$ with time for the GW strain amplitude
in row 2 of Table 1; $\omega=$ 0.1 $Hz$, while the separations of the angular
arms are here set to 0.001$"$, instead of 0.01$"$, thus generating variations
of a few $\mu$as. Therefore, this figure and the considerations on the cases
presented in Table 1, confirm that an astrometric antenna, capable of
monitoring periodic signals of amplitude $>$ 1 $\mu$as, possesses the ability
of measuring GW’s associated with a range of coalescing massive BH’s events.

Finally, row 4 of Table I refers to smaller amplitudes as expected for GW’s
from core collapse events (at $\omega\sim 100Hz$), or, at much shorter
frequencies, from resolved and unresolved MW binaries. These cases appear
beyond today’s technology on current missions, including payloads that will be
flying into orbit within the next few years.

Expanding the access to the physics of GW’s for an astrometric antenna would
then require dedicated technological developments (see e.g. Riva et al., 2010;
Lattanzi, 2012). Regardless of the actual detectability the priority of this
article is to prove the concept of a novel idea that might develop into new
space-borne astrometric instrumentation for the study of GW’s. From this
perspective, it is evident that the angular arms of the 3-LOS GW antenna, as
schematically depicted in Fig. 1 (a), would register the astrometric signals
with different amplitudes and phases depending on actual orientations. For the
examples in Fig.s 1 (b) and 1 (c), these data would immediately tell the
direction of arrival of the gravitation strain causing the trembling in the
antenna arms (the chosen stellar pairs) as that of minimum signal. Therefore,
the ultimate precision with which a direction can be recovered is sub-$mas$,
as the sources will be in the Gaia Catalogs (see, e.g., Lindegren et al.,
2021). Or, one could say the actual directional uncertainty is comparable to
the angles separating the stellar pairs used as arms, i.e., $\sim$ 10 mas or
less. Either case, these are unprecedented numbers and pinpointing source
directions with such accuracies would tremendously help multi-wavelength and
multi-messenger follow-up investigations.

##  IV Concluding remarks and perspectives

Immediately after the initial successful efforts at the idea of an antenna for
measuring GW’s using astrometry from a telescope in space (see sec. 6.4 in
Crosta, 2019), we initiated efforts to prove, with simulations and laboratory
tests, the feasibility of an astrometric antenna based on the precepts
described in this article (Gai et al., 2020a, b; Riva et al., 2020, and
references therein). We refer to those studies for results and discussions on
what can be currently said on implementation issues like, just to mention a
few: i) the very possibility to build a 3-LOS multiplexing telescope (Sun et
al., 2021, and references therein), ii) the limit of centering accuracies of
star-like images on digital detectors, iii) actual (beyond Rayleigh’s)
resolution limits for the antenna arms (depending not only on optics and
detection system, but also on magnitude and color of the stellar pairs), iv)
other natural (intrinsic or cosmic) causes of astrometric noise as stellar
variability (both astrometric and photometric), and v) identify (via spatial
laser metrology of critical degrees-of-freedom) and deal with instrumental
noise mimicing unwanted variations of the antenna arms $\psi_{i_{0},j_{0}}$.
In addition, we will have to simulate much more realistic scenarios (i.e.,
more general forms of GW’s and use real-sky pairs) and conditions (realistic
noise levels) to investigate viable strategies for the actual retrieval of
amplitude and phase (carrying the direction information). However, for these
particular aspects we can certainly draw from the great amount of work done,
and proven on real data by the LIGO and VIRGO collaborations.

To date, one generally considers the global reduction process at the end of a
mission like Gaia to extract the GW astrometric signals. In such a context,
looking for variation of the position of a single source on the sky at nas
level implies the knowledge of an ”absolute” reference, e.g. the telescope
LOS, at comparable precision $\sim 10^{-15}$. This is an unprecedented
requirement on the reconstructed attitude of a science satellite. The proposed
differential technique in Gai et al. (2020a, b) aims at nas measurement over a
distance of the order of 1”, thus implicitly reducing the relative precision
requirement to $\sim 10^{-9}$, with an improvement of six orders of magnitude.
Also, instrument calibration requirements are strongly alleviated. In Gaia, we
have a variation of the electro-optical response of hundreds of mas over the
$0.5^{\circ}$ field, calibrated to the $\mu$as level. Assuming a linear model,
the corresponding electro-optical response variation for an astrometric GW
telescope over 1” would be in the range of hundreds of $\mu$as, i.e. a
comparable calibration ”power” would be required to scale the measurement
reliability to the nas regime. Actually, optimal telescope design, exploiting
the higher-than-linear decrease of many aberrations close to the optical axis,
would significantly reduce such instrumental contribution (Riva et al., 2020).

Nevertheless, as we show in this work, the astrometric observable could
amplify the GW-induced signal if one takes into account the angle between two
space-like directions of light (Crosta, 2019) in the framework of general
relativistic astrometric models (Crosta et al., 2017). In such a case one is
exempted from dealing with the satellite’s attitude and the GW astrometric
measurement can be translated into an observation equation accounting for a
wide range of frequencies.

The diversity of GW frequencies potentially included in Eq.(9) and the
appropriate modeling of the gravitational shift could improve PTA and/or LISA
observations (Verbiest et al., 2016; Amaro-Seoane et al., 2017) and the low
frequency domain due to periodic sources (e.g., Galactic binary WDs identified
by Gaia), further enhancing the mapping of the Milky Way substructures
(Wilhelm et al., 2021). Moreover, the advantage of Eq. (6) is to exploit a
large number of null geodesics, so to better scrutinize the GW direction (a
critical aspect of the GW detection). This same feature will also enable tests
on GW polarization modes by combining different telescope orientations.
In conclusion, the potential of the relativistic astrometric observable
advocated here for the astrometric detection and identification of
gravitational waves using solely stars will push on technological advances and
will significantly add to the best GW detection procedures. In fact, given
that stars allow us to exploit a huge number of configurations, we could
potentially monitor gravitational waves coming from any direction and, at the
same time, link the properties of a GW source with extensive statistics. Each
GW mechanism has typical signatures with its own associated uncertainties
that, if not instrumental, could identify the underlying physics of the
corresponding GW production. This helps in a truly complementary way all of
the efforts dedicated to the multiband GW searches bridging low and high
frequencies at different redshifts (see, for example, Sedda et al., 2020;
Aggarwal et al., 2011; Burke–Spolaor et al., 2019). When the two unperturbed
line-of-sights, although angularly very close, are related to different
stellar distances, it would be also possible to investigate time retarded
effects and test the GW speed. In addition, suitable choice of the strain
$h^{GW}$ could pave the way for new GW tests on the gravitation interaction
with photons. It would suffice modeling the $F^{GW}$ function accordingly.

## Appendix A: The direction cosine for the GW observation equation

The direction cosine of two light directions is defined as:

$\cos\psi_{1,2}=\frac{P(u)_{\alpha\beta}k^{\alpha}_{1}k^{\beta}_{2}}{(\sqrt{P(u)_{\alpha\beta}k^{\alpha}_{1}k^{\beta}_{1}})(\sqrt{P(u)_{\alpha\beta}k^{\alpha}_{2}k^{\beta}_{2}})}$
(10)

where $P(u)_{\alpha\beta}=g_{\alpha\beta}+u_{\alpha}u_{\beta}$ is the operator
that projects with respect to the local barycentric observer $u^{\alpha}$.
Then the photon 4-momentum can be decomposed as

$k^{\alpha}=-(u|k)u^{\alpha}+\ell^{\alpha}\,,$ (11)

where $(u|k)=g_{\alpha\beta}u^{\alpha}k^{\beta}$ and $\ell^{\alpha}$ is the
spatial null vector projected on the rest space of $u^{\alpha}$. Defining

$\bar{k}^{\alpha}=-\frac{k^{\alpha}}{(u|k)}\,,\,\,\,\,\bar{\ell}^{\alpha}=-\frac{\ell^{\alpha}}{(u|k)}=\bar{k}^{\alpha}-u^{\alpha}\,.$
(12)

the direction cosine simplifies as

$\cos\psi_{1,2}={g}_{\alpha\beta}({\bar{\ell}}^{\alpha}_{1}{\bar{\ell}}^{\beta}_{2})_{obs}.$
(13)

The BCRS metric is defined by IAU resolutions as a post-Newtonian (pN)
solution of the Einstein field equations. Thus, one has to take into account
terms of this metric accurate to the order of the GW perturbations sought for.
Dropping the sum symbol, let us express the metric including both sources as:

${g}_{\alpha\beta}=\eta_{\alpha\beta}+\epsilon{h}^{SS}_{(1)\alpha\beta}+\epsilon^{2}{h}^{SS}_{(2)\alpha\beta}+\epsilon^{3}{h}^{SS}_{(3)\alpha\beta}+\epsilon^{4}{h}^{SS}_{(4)\alpha\beta}+{h}_{\alpha\beta}^{GW}+O(\epsilon^{5})$
(14)

where $\epsilon$ is of the order of $v/c$, being $v$ the typical velocity of
each relevant SS metric source, and the subscripts in parenthesis indicate the
order of approximation. We assume that the GW perturbations are of order
$\epsilon^{4}$ at best, i.e., at the nanoarcsecond level. Like for the SS
metric, the SS contribution to light direction can be approximated as:

${\bar{\ell}}^{\alpha(SS)}_{obs}={\bar{\ell}}^{\alpha}_{{}_{0}}+\epsilon{\bar{\ell}}^{\alpha}_{{}_{(1)}}+\epsilon^{2}{\bar{\ell}}^{\alpha}_{{}_{(2)}}+\epsilon^{3}{\bar{\ell}}^{\alpha}_{{}_{(3)}}+\epsilon^{4}{\bar{\ell}}^{\alpha}_{{}_{(4)}}+O(\epsilon^{5}).$
(15)

From the assumptions above, one finally finds:

$\cos\psi_{1,2}=\cos\psi_{1,2}^{SS}+\eta_{\alpha\beta}({\bar{\ell}}^{\alpha}_{1_{0}}\delta\ell_{2}^{\beta}+{\bar{\ell}}^{\alpha}_{2_{0}}\delta\ell_{1}^{\beta})_{obs}+h^{GW}_{\alpha\beta}{\bar{\ell}}^{\alpha}_{1_{0}}{\bar{\ell}}^{\beta}_{2_{0}}+O(\epsilon^{5}).$
(16)

The same can be obtained by differentiating equation (16), much like the
linearization procedure utilized in the Gaia mission (Crosta et al., 2017) to
solve for the unknown astrometric parameters of each individual direction, but
this time with the GW strain as unknown:

$-\sin(\psi_{1,2})d\psi_{1,2}=-\sin(\psi_{1,2}^{SS})d\psi_{1,2}^{SS}+dF^{GW}_{1,2}+O(\epsilon^{5}),$
(17)

that, for $\delta\psi^{GW}_{1,2}\ll 1$ and
$\psi_{1,2}=\psi_{1,2}^{SS}+\delta\psi^{GW}_{1,2}$, can be approximated as

$-\sin(\psi_{1,2}^{SS})(d\psi_{1,2}^{SS}+d(\delta\psi_{1,2}^{GW}))\approx-\sin(\psi_{1,2}^{SS})d\psi_{1,2}^{SS}+dF^{GW}_{1,2},$
(18)

or, better, as

$d(\delta\psi_{1,2}^{GW})=-\frac{dF^{GW}_{1,2}}{\sin(\psi_{1,2}^{SS})}+O(\epsilon^{5}).$
(19)

where ${\bar{\ell}}^{\alpha}_{obs}$ is the local line-of-sight at the
observation time.

## Appendix B: The total direction cosine versus an additive GW direction
cosine

In this section we clarify the assertions made in the Introduction on the
consequences, when in the presence of a passing GW, of utilizing the direction
cosine observable by simply extending what it is done in the context of the
Gaia mission, where the direction cosine refers to the angle of the incoming
light to the observer attitude-tetrad $E^{\alpha}_{\hat{a}}$.

Let us consider the $\it{a~{}priori}$ assumption that the effect of a GW is
that of adding a cosine term to the direction cosine associated with the SS
metric, i.e., the total (tot) direction cosine is given by:

$\cos(\psi)^{SS}+\cos(\psi)^{GW}.$ (20)

With the extra assumption that the correction $\delta\bar{\ell}_{(GW)}$,
induced by the GW to the SS line-of-sight at the observer $\bar{\ell}_{(SS)}$,
is (formally) known, Eq. (20) would read:

$(\eta_{\mu\nu}+h^{SS}_{\mu\nu})(\bar{\ell}^{\mu}_{(SS)}E^{\nu}_{\hat{a}})+(\eta_{\mu\nu}+h^{GW}_{\mu\nu})(\delta\bar{\ell}^{\mu}_{(GW)}E^{\nu}_{\hat{a}})\approx\cos(\psi)^{SS}+\eta_{\mu\nu}\delta\bar{\ell}^{\mu}_{(GW)}E^{\nu}_{\hat{a}},$
(21)

implying no explicit dependence on the strain $h_{GW}$.

If, on the other hand, the GW perturbation is not a known part of the observed
direction at the observer, one would have:

$\displaystyle(\eta_{\mu\nu}+h^{SS}_{\mu\nu})[(\bar{\ell}^{\mu}+\delta\bar{\ell}^{\mu})E^{\nu}_{\hat{a}}]+(\eta_{\mu\nu}+h^{GW}_{\mu\nu})[(\bar{\ell}^{\mu}+\delta\bar{\ell}^{\mu})E^{\nu}_{\hat{a}}]$
(22) $\displaystyle\approx$
$\displaystyle\cos(\psi)^{SS}+2\eta_{\mu\nu}\delta\bar{\ell}^{\mu}E^{\nu}_{\hat{a}}+\eta_{\mu\nu}\bar{\ell}^{\mu}E^{\nu}_{\hat{a}}+h^{GW}_{\mu\nu}\bar{\ell}^{\mu}_{\not
0}E^{\nu}_{\hat{a}},$

with the flat Minkowskian metric term entering twice in the Equation for the
$\bar{\ell}^{\alpha}$ component. Then, the simple addition to the ”background”
direction cosine in the SS metric (i.e. BCRS would introduce the flat
Minkowskian contribution twice without a priori disentanglement from the
local-line-of-sight of the GW component or, if the GW shift is considered as a
separate part, it would imply to discard a priori the $h^{GW}$ strain in the
observation equation.

It is only when we drop the assumption made with Eq. (20), i.e. when working
directly with the total direction cosine, that we finally recover an
expression similar to observation equation in the Main article as:

$\cos(\psi)_{tot}=(\eta_{\mu\nu}+h^{SS}_{\mu\nu}+h^{GW}_{\mu\nu})[(\bar{\ell}^{\mu}+\delta\bar{\ell}^{\mu})E^{\nu}_{\hat{a}}]\approx\cos(\psi)^{SS}+\eta_{\mu\nu}\delta\bar{\ell}^{\mu}E^{\nu}_{\hat{a}}+h^{GW}_{\mu\nu}\bar{\ell}^{\mu}E^{\nu}_{\hat{a}}.$
(23)

However, this form of the observation equation explicitly depends on the
observer orientation, i.e., on satellite attitude, with the drawbacks
mentioned in the Main article in relation to attitude errors heavily affecting
the error budget of the measurements.

## Appendix C: The astrometric gravitational shift

The four tangent vector to a null geodesic satisfies the well known
conditions:

$k^{\alpha}\nabla_{\alpha}k^{\beta}=0\,,\qquad k^{\alpha}k_{\alpha}=0\,,$ (24)

$\nabla_{\alpha}$ being the covariant derivative associated with the spacetime
metric. The decomposition of the photon 4-momentum with respect to an observer
$u^{\alpha}$ implies that the trajectory is parametrized by $\sigma$ such that

$\bar{k}^{\alpha}=-\frac{k^{\alpha}}{(u|k)}=\frac{{\rm d}x^{\alpha}}{{\rm
d}\sigma},$ (25)

and equation (24) becomes

$\bar{k}^{\alpha}\nabla_{\alpha}\bar{k}^{\beta}=-\frac{{\rm
d}\ln{[-(u|k)]}}{{\rm d}\sigma}\bar{k}^{\beta}\,,$ (26)

which is related to the affine parameter $\lambda$ by ${\rm d}\sigma=-(u|k)\rm
d\lambda$.

It is easy to check that in the case of a static observer:

$\frac{{\rm d}\ln{[-(u|k)]}}{{\rm
d}\sigma}=\bar{\ell}^{\alpha}\bar{\ell}^{\beta}K_{\alpha\beta}-\bar{\ell}^{\alpha}a_{\alpha}=-\bar{\ell}^{\alpha}\bar{\ell}^{\beta}\theta_{\alpha\beta}-\bar{\ell}^{\alpha}a_{\alpha}\,,$
(27)

where the two spatial fields coming from the splitting of the covariant
derivative of $u$, i.e.,
$\nabla_{\beta}u^{\alpha}=-a^{\alpha}u_{\beta}-K^{\alpha}{}_{\beta}$, the
acceleration vector $a^{\alpha}$ and the kinematical tensor
$K^{\alpha}{}_{\beta}=\omega^{\alpha}{}_{\beta}-\theta^{\alpha}{}_{\beta}$ are
expressed as a combination of the vorticity and expansions of the congruence
of curves related to fiducial observers $u^{\alpha}$. Thus, the geodesic
equation transforms into

$\frac{{\rm d}\bar{k}^{\alpha}}{{\rm
d}\sigma}+\Gamma^{\alpha}{}_{\mu\nu}\bar{k}^{\mu}\bar{k}^{\nu}-\left[\bar{\ell}^{\mu}\bar{\ell}^{\nu}\theta_{\mu\nu}+\bar{\ell}^{\mu}a_{\mu}\right]\bar{k}^{\alpha}=0\,,$
(28)

or

$\displaystyle\frac{{\rm d}\bar{\ell}^{\alpha}}{{\rm d}\sigma}$
$\displaystyle+$
$\displaystyle\Gamma^{\alpha}{}_{\mu\nu}\bar{\ell}^{\mu}(\bar{\ell}^{\nu}+u^{\nu})+a^{\alpha}-K^{\alpha}{}_{\sigma}\bar{\ell}^{\sigma}-\left[\bar{\ell}^{\mu}\bar{\ell}^{\nu}\theta_{\mu\nu}+\bar{\ell}^{\mu}a_{\mu}\right](\bar{\ell}^{\alpha}+u^{\alpha})=0\,,$
(29)

for the unknown local line-of-sight $\bar{\ell}^{\alpha}$. In case of static
observers, the kinematical fields reduce to

$a^{i}=\partial_{0}h_{0i}-\frac{1}{2}\partial_{i}h_{00}=h_{0i,0}-\frac{1}{2}h_{00,i}\,,\qquad\theta_{ij}=\frac{1}{2}h_{ij,0}\,,\qquad\omega_{ij}=-h_{0[i,j]}\,.$
(30)

Note that considering the metric as an approximate solution of the Einstein
field equation composed of a background part plus a GW perturbation, also the
affine coefficients can be split respectively into two parts:

$\Gamma^{\alpha}{}_{\mu\nu}=\frac{1}{2}g^{\alpha\rho}\left(g_{\rho\beta,\gamma}+g_{\rho\gamma,\beta}-g_{\beta\gamma,\rho}\right)\approx\Gamma^{\alpha(SS)}_{\beta\gamma}+\Gamma^{\alpha(GW)}_{\beta\gamma}\,,$
(31)

at the first order of the perturbation. The same for the parameter sigma:

$d\sigma=-(u|k)d\lambda=-[(g_{\alpha\beta}^{(SS)}+h_{\alpha\beta}^{(GW)})\,u^{\alpha}k^{\beta}]d\lambda.$
(32)

The TT gauge choice implies that the second term of equation (32) does not
contribute.

All of the above implies the possibility again to split equation (29) into the
Solar System part plus the GW one, thus it allows to integrate separately each
term . As a matter of fact, at the order of $\epsilon^{4}$, it is possible to
isolate from equation (29) the contribution of the GW part and obtain

$\frac{d\delta\ell^{i}}{d\sigma}\approx-\frac{1}{2}\bar{\ell}^{j}_{{}_{0}}\bar{\ell}^{k}_{{}_{0}}(2h^{GW}_{ij,k}-h^{GW}_{jk,i})-\bar{\ell}^{j}_{{}_{0}}h^{GW}_{ij,0}+\frac{\bar{\ell}^{j}_{{}_{0}}\bar{\ell}^{k}_{{}_{0}}\bar{\ell}^{i}_{{}_{0}}}{2}h^{GW}_{jk,0}.$
(33)

Since $d\sigma=d\lambda+O(\epsilon^{2})$ and assuming for the photon
trajectory $x^{0}(\sigma)=x^{0}_{obs}+\sigma+O(\epsilon^{2})$ and
$x^{i}(\sigma)=x^{i}_{obs}+\ell^{i}_{{}_{0}}\sigma+O(\epsilon^{2})$, the
argument of $h_{ij}$ becomes:

$\tilde{k}_{\alpha}x^{\alpha}=\tilde{k}_{0}\sigma(1+p\cdot\bar{\ell}_{{}_{0}})+\tilde{\psi}+O(\epsilon^{2})$
(34)

where $\tilde{\psi}=\tilde{k}_{\alpha}x_{obs}^{\alpha}$ can be considered a
phase term. Then,

$h_{ij,0}=\frac{1}{(1+p\cdot\bar{\ell}_{{}_{0}})}\frac{dh}{d\sigma},\,\,\,\,h_{ij,k}=\frac{p^{k}}{(1+p\cdot\bar{\ell}_{{}_{0}})}\frac{dh}{d\sigma}.$
(35)

Via a direct integration of equation (33) we easily obtain the gravitational
shift of the local direction:

$\int^{\sigma_{*}}_{\sigma_{obs}}d\delta\ell^{i}=\frac{\bar{\ell}^{i}_{{}_{0}}+p^{i}}{2(1+p\cdot\bar{\ell}_{{}_{0}})}\bar{\ell}^{j}_{{}_{0}}\bar{\ell}^{k}_{{}_{0}}\int^{\sigma_{*}}_{\sigma_{obs}}dh_{jk}^{(GW)}(\sigma)-\frac{\bar{\ell}^{j}_{{}_{0}}}{2}\int^{\sigma_{*}}_{\sigma_{obs}}dh_{ij}^{(GW)}(\sigma)+O(\epsilon^{5}),$
(36)

which coincides with the result in Book & Flanagan (2011) when the distance to
the stellar source is many gravitational waves away, namely the detection
occurs in the far-away wave zone.

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