# Solar System-scale interferometry on fast radio bursts could measure cosmic
distances with sub-percent precision

Kyle Boone<EMAIL_ADDRESS>Department of Astronomy, University of Washington,
3910 15th Ave NE, Seattle, WA 98195, USA  Matthew McQuinn<EMAIL_ADDRESS>Department of Astronomy, University of Washington, 3910 15th Ave NE, Seattle,
WA 98195, USA

###### Abstract

The light from an extragalactic source at a distance $d$ will arrive at
detectors separated by $100~{}$AU at times that differ by as much as
$120\,(d/100\,{\rm Mpc})^{-1}~{}$nanoseconds because of the curvature of the
wave front. At gigahertz frequencies, the arrival time difference of a point
source can be determined to better than a nanosecond with interferometry. If
the space-time positions of the detectors are known to a few centimeters,
comparable to the accuracy to which very long baseline interferometry
baselines and global navigation satellite systems (GNSS) geolocations are
constrained, nanosecond timing would allow competitive cosmological
constraints. We show that a four-detector constellation at Solar radii of
$\gtrsim 10\,$AU could measure geometric distances to individual sources with
sub-percent precision. The precision increases quadratically with baseline
length. Fast radio bursts (FRBs) are the only known bright extragalactic radio
source that are sufficiently point-like for this experiment, and the simplest
approach would target the population of repeating FRBs. Galactic scattering
limits the timing precision at $\lesssim 3\,$GHz, whereas at higher
frequencies the precision is set by removing the differential dispersion
between the detectors. Furthermore, for baselines greater than $100~{}$AU,
Shapiro time delays limit the precision, but their effect can be cleaned at
the cost of two additional detectors. Outer Solar System accelerations that
result in $\sim$1 cm uncertainty in detector positions could be corrected for
with weekly GNSS-like trilaterations between members of the constellation. The
proposed interferometer would not only provide a geometric constraint on the
Hubble constant, but also could advance Solar System, pulsar, and
gravitational wave science.

cosmological parameters from LSS, fast radio bursts, radio astronomy, dark
energy experiments

††software:  Astropy (Astropy Collaboration et al., 2013, 2018), emcee
(Foreman-Mackey et al., 2013), Matplotlib (Hunter, 2007), NumPy (Harris et
al., 2020),

##  1 Introduction

Fast radio bursts (FRBs) are millisecond radio transients that are generally
of extragalactic origin (Lorimer et al., 2007; Thornton et al., 2013;
Tendulkar et al., 2017; Bannister et al., 2019; Cordes & Chatterjee, 2019;
Petroff et al., 2022). Hundreds of FRBs have been discovered to date, tens of
which have been found to be repeating, with $\sim 10^{4}$ on the sky per day
above $1\;$Jy ms – a detectable fluence for many radio telescopes (Spitler et
al., 2016; CHIME/FRB Collaboration et al., 2021). Not only are FRBs
interesting for identifying and studying the extreme radiative processes that
create them, likely associated with magnetars (Bochenek et al., 2020;
CHIME/FRB Collaboration et al., 2020), but propagation effects that alter the
received electromagnetic waves can be used as a tool to study the missing
baryon problem, the circumgalactic media of galaxies, and the cosmic
reionization history (McQuinn, 2014; Ravi, 2019; Prochaska et al., 2019;
Beniamini et al., 2021; Heimersheim et al., 2021).

Several studies have also identified potential ways to use FRBs for precision
cosmology. The simplest of which is to use that the frequency dependence of
the wave front arrival to infer the column of electrons along the sightline
and create a Hubble diagram-like electron column versus redshift relation.
While it is unlikely that the host galaxy electron column will be small enough
to allow a precise determination of this relation (Macquart et al., 2020), a
way around this uncertainty is to instead use that correlations between
sightlines are only sensitive to the cosmological density fluctuations (Masui
& Sigurdson, 2015). This idea would map the fluctuating electron density to
constrain cosmological parameters, somewhat analogous to a weak lensing
survey.

An alternative route to cosmological constraints uses a key attribute of FRBs
– that their arrival time can be measured very accurately. The point-like
nature of FRBs allows coherent timing with precision of better than the
inverse of their frequency, a nanosecond at a gigahertz. FRBs could be timed
much more precisely than the slowly varying quasars that are used in current
strong lensing time delay analyses, which are starting to put competitive
constraints on the Hubble constant (Li et al., 2018). Unfortunately,
uncertainties in the mass modeling of the lens system (Kochanek, 2021) as well
as in the projected mass (Bar-Kana, 1996) may prevent percent-level cosmology
with lensing time delays regardless of the timing precision. One potential way
around these limitations involves lensed, repeating FRBs. Measuring the $\sim
10^{-3}$ s yr-1 evolution in the time between lensed images due to the
evolving redshift of the lens and source would constrain the cosmology (Zitrin
& Eichler, 2018; Wucknitz et al., 2021), although this rate is comparable to
that from the mass assembly of the lens.

In this article, we propose a new geometric approach to constrain cosmology
with FRBs. The wave front of an FRB will have a small curvature when it
reaches our Solar System. By measuring the arrival time of the same FRB at
four separate radiometers, this curvature can be measured directly and used to
infer the distance. (A similar idea has been applied to pulsar timing arrays
in D’Orazio & Loeb 2021 and McGrath et al. 2022.) An illustration of this idea
is shown in Figure 1. We will show that a constellation of detectors separated
by 100 AU and observing at $\gtrsim 3$ GHz would be able to measure the
cosmological distance to an FRB at 100 Mpc with an accuracy of $\sim$0.1%.
Sub-percent distances translate to sub-percent constraints on cosmological
parameters via the redshift–distance relation once sufficiently into the
Hubble flow. This idea also has the appeal of being a geometric measurement of
distance in cosmology akin to parallax, not requiring the assumptions of other
distance methods (e.g. the standardization of Type Ia supernovae; Phillips
1993; or the $\Lambda$CDM model for cosmological density fluctuations;
Dodelson 2003).

Figure 1: Example of a detector configuration that can be used to measure the
distance to a source from the curvature of its wave front. The signal will
arrive at detector B before it is seen at detectors A or C. By comparing the
arrival times at the three detectors we can infer the distance to the source.
Note that we can only measure the difference in arrival times, not the
distances $d_{i}$ directly. With two detectors the distance to the source is
degenerate with the angular position on the sky $\theta$. With three detectors
in two dimensions, or four detectors in three dimensions, this degeneracy is
broken and the distance to the source can be inferred.

The proposed experiment requires measuring the arrival times of FRBs with sub-
nanosecond precision for Solar System-sized baselines. Sub-nanosecond timing
is regularly done with very long baseline interferometry (VLBI; Schuh &
Behrend 2012). With Earth-scale baselines, VLBI timing has allowed measuring
the angular coordinates of thousands of radio sources with milli-arcsecond
errors (Beasley et al. 2002; and $\sim 10\,\mu$-arcsecond for Sagittarius A∗
and the black hole at the center of M87; Event Horizon Telescope Collaboration
et al. 2019). VLBI has also been done in space, with the 10 m dish on the
Spektr-R satellite as part of the RadioAstron program successfully finding
correlations with a 30 Earth radius baseline (Hirabayashi et al., 1998;
Kardashev et al., 2013). VLBI is now regularly being done on FRBs (Marcote et
al., 2017, 2020; Kirsten et al., 2022).

We also require the detectors to be localized with an accuracy of several
centimeters over Solar System scales. Modern positioning systems can localize
global navigation satellite systems (GNSS; such as the United State’s GPS,
Russia’s GLONASS, China’s BeiDou and Europe’s Galileo) to meter or even
centimeter precision (e.g. Misra & Enge, 2012). Millimeter accuracies on
distance measurements are obtained with optimizations (longer integrations,
having well-localized places on Earth for a differential measurement) that are
able to break the integer wavelength ambiguity in the carrier phase. Using
very similar methods to GNSS systems, precision ranging is regularly performed
to NASA satellites across the Solar System by the Deep Space
Network,111https://www.nasa.gov/directorates/heo/scan/services/networks/deep_space_network/
reaching several meter precision in the separation and with the potential for
improvement (Border & Paik, 2009). The Deep Space Network has executed precise
radio ranging to the Pioneer, Voyager and New Horizons missions to the outer
Solar System.

While placing radio telescopes on different sides of the Solar System would
require a significant effort, a further motivation is that cosmology is near
the end of how well we can measure parameters with traditional techniques.
Much of the useful cosmic volume for constraining dark energy will soon be
mapped, and known techniques for constraining cosmological parameters cannot
achieve errors below $\sim 1$% on most parameters. Currently there are 10%
tensions in the Hubble parameter determined with the most accepted techniques,
namely the cosmic microwave background and Type Ia supernovae calibrated with
parallaxes and then Cepheid variable stars (Riess et al., 2019; Di Valentino
et al., 2021). If future measurements indicate a few-percent deviation from
the $\Lambda$CDM expectation, would the community believe it? The proposed
experiment offers a potential way forward, enabling geometric sub-percent
distance measurements where the major sources of error appear to be under the
instrumentalist’s control. We show that a _single_ FRB at $d=200\,$Mpc could
be used to make a 1% measurement of the Hubble constant for radio telescopes
at Solar radii of $\gtrsim 20~{}$AU, with the 1% set by the uncertainty in the
peculiar velocity of the host galaxy. This measurement is comparable to the
SHOES 1% measurement of the Hubble constant using $\approx 100$ Type Ia
supernovae with a median distance of $d=200\,$Mpc (Riess et al., 2022). Each
supernova provides a $10\%$ distance measurement, and so $\approx 100$
supernovae are required for 1% precision; similar averaging can be done in our
proposed experiment to achieve sub-percent measurements of the Hubble
constant. We further show that $\sim 100~{}$AU baselines could achieve 1%
constraints on a single FRB out to $d=3000\,$Mpc or $z=1$.

Perhaps the most apt comparison is with gravitational wave standard sirens.
Forecasts for a proposed next-generation instrument, the Einstein Telescope,
are $\sim 1$% distance errors to a low redshift and $\sim 5\%$ percent to
$z=1$ (Maggiore et al., 2020), with these predictions reliant on assumptions
about source demographics and the identification of electromagnetic
counterparts. Our proposal offers the potential for higher precision even from
a single FRB.

There are several applications beyond the obvious Hubble constant and dark
energy science. Another cosmological parameter for which it would be
interesting to push beyond current constraints is the spatial curvature. If
slow-roll inflation starts with a comoving Hubble radius that is comparable to
our present-day horizon, the Universe could be more curved than the part in
$10^{5}$ inflationary expectation, and there might be anthropic reasons for
why we inhabit a region with an enhanced negative spatial curvature (Freivogel
et al., 2006). Additionally, the proposed extremely long baseline instrument
could potentially measure the mass distribution in the outer Solar System, the
distance to Galactic pulsars, constrain clumpy dark matter models, resolve the
radio emission from pulsars, more directly measure the density fluctuation
spectrum of ISM turbulence, and reach interesting sensitivities to
$<100\,\mu$Hz gravitational waves.

This paper is organized as follows. We present the idea for distance
constraints from wave front curvature in § 2, and we argue that FRBs are
likely the only extragalactic source that is sufficiently point-like for this
method in § 3. We then show that various systematics, namely ISM scattering,
dispersion, and gravitational time delays, appear to be under control for
$\nu\gtrsim 3\;$GHz, with gravitational time delays limiting the precision for
$\gtrsim 100\;$AU antenna separations (§ 4; but also fully removable with two
more detectors). We discuss the feasibility of calibrating the antennas’
space-time locations to within one wavelength in § 6. Finally, § 7 synthesizes
our estimates for the timing noise to forecast the achievable cosmic distance
constraints, and § 8 highlights other sciences applications that could be
addressed with Solar System-scale interferometry. Appendix A generalizes our
calculations to curved space, and the other appendixes add details to some of
the systematics calculations.

##  2 Geometric distance measurements with time delays

Here we describe how direct cosmological distances can be ascertained using
the time delays between radiometers. This section presents a Euclidean-space
derivation, which gives the essential idea and is a good approximation for
sources that are not at appreciable redshifts. We generalize this derivation
to Friedman-Robertson-Walker (FRW) space-times in Appendix A. In flat FRW
space-times, the distances in the Euclidean derivation are simply mapped to
the comoving light-travel distances such that the following carries over
directly.

Let us first consider the case of two radiometers. Without loss of generality,
we label one of our detectors the “reference detector,” and we define a
coordinate system so that this detector is at the origin. We label the
position of the source as ${\boldsymbol{d}}$, the position of a second
detector as ${\boldsymbol{x}}$, and the vector between the second detector and
the source as ${\boldsymbol{d}}_{x}$. Symbols in bold characters refer to
vectors, while symbols in regular fonts refer to the lengths of the
corresponding vectors. An illustration of this configuration is shown in
Figure 2.

Figure 2: The triangular configuration of two detectors and the source used
for timing.

The distance between the second detector and the source can be written as

$\displaystyle d_{x}$
$\displaystyle=|{\boldsymbol{d}}-{\boldsymbol{x}}|=\sqrt{d^{2}+x^{2}-2{\boldsymbol{x}}\cdot{\boldsymbol{d}}},$
(1)

Since $x\ll d$ (i.e. the distance between the two detectors is much smaller
than the distance to the source), we expand the previous equation in $x/d$,
yielding

$d_{x}=d-{\boldsymbol{x}}\cdot{\boldsymbol{\widehat{d}}}+\frac{1}{2d}\left(x^{2}-({\boldsymbol{x}}\cdot{\boldsymbol{\widehat{d}}})^{2}\right)+\mathcal{O}(x^{3}/d^{2}),$
(2)

where ${\boldsymbol{\widehat{d}}}$ is the unit vector in the direction of
${\boldsymbol{d}}$.

The arrival time of the signal at the second detector relative to the first
one is then given by:

$\displaystyle c\Delta t$
$\displaystyle=d_{x}-d=-{\boldsymbol{x}}\cdot{\boldsymbol{\widehat{d}}}+\underbrace{\frac{1}{2\,d}\left(x^{2}-({\boldsymbol{x}}\cdot{\boldsymbol{\widehat{d}}})^{2}\right)}_{c\Delta
t_{d}}+\mathcal{O}\left(\frac{x^{3}}{d^{2}}\right).$ (3)

The $-{\boldsymbol{x}}\cdot{\boldsymbol{\widehat{d}}}$ term is the difference
in time for a flat wave front to travel past each of the detectors. The second
term in the equation labeled $\Delta t_{d}$ is the “distance-dependent time
delay” and is quadratic in $x$ due to the fact that the wave front is not flat
when the source is at a finite distance.

Assuming that the detector geometry is known to better than the timing
precision (the details of which will be discussed in Section 6), this equation
has three unknown variables: two for the angular position of the source on the
sky and one for the distance to the source. With at least four detectors
measuring three independent time delays this system is fully constrained and
can be used to infer the distance to the source.222Another way of doing the
constraint counting is to note that four detectors yield the four equations:
$|{\boldsymbol{d}}-{\boldsymbol{x}}_{i}|^{2}=c^{2}(t_{\rm FRB}-t_{i})^{2}$,
where $i$ indexes each detector. The time of the FRB $t_{\rm FRB}$ is one
unknown and so this leaves just enough constraints to measure
${\boldsymbol{d}}$, assuming that the space-time positions of the detectors
are known. This is not unlike the four satellites for precise GNSS
positioning; there the fourth unknown is the signal arrival time at the
receiver, as in the typical setup the receiver does not have a high precision
clock. The exact accuracy of this measurement depends on the full detector
configuration. For typical configurations, we can approximate the fractional
distance uncertainty $\sigma_{d}/d$ as

$\displaystyle\frac{\sigma_{d}}{d}\sim\frac{\sigma_{t}}{\Delta t_{d}}$ (4)

where $\sigma_{t}$ is the accuracy of a time delay measurement on a given
detector pair. This expression was confirmed experimentally using Fisher
matrix calculations as well as a Markov Chain Monte Carlo (MCMC)
implementation in the emcee package (Foreman-Mackey et al., 2013). For a range
of different four-detector configurations where the detectors are all
separated by similar projected baselines, we find that the error is
approximately $\Delta t_{d}$. The errors quickly become smaller for more than
four detectors owing to the large number of detector pairs, scaling roughly as
$\propto 1/\sqrt{(N_{A}-n_{p})(N_{A}-n_{p}-1)/2}$ where $N_{A}$ is the number
of detectors and $n_{p}$ is the number of additional parameters, being
$n_{p}=2$ for the minimal scenario considered in this section where the
additional parameters are the source angular coordinates – the scenario that
we find applies until the baselines surpass 100 AU (§ 7).

The distance-dependent time delay $\Delta t_{d}$ can be rewritten as

$\Delta
t_{d}=\frac{x_{\perp}^{2}}{2\,c\,d}=120~{}\text{ns}~{}\left(\frac{x_{\perp}}{100~{}\text{AU}}\right)^{2}\left(\frac{d}{100~{}\text{Mpc}}\right)^{-1},$
(5)

where $1\;$AU is the Earth-Sun distance and

${\boldsymbol{x}}_{\perp}\equiv{\boldsymbol{x}}-({\boldsymbol{x}}\cdot{\boldsymbol{\widehat{d}}}){\boldsymbol{\widehat{d}}}$

is the projection of the detector separation ${\boldsymbol{x}}$ that is
perpendicular to the line of sight.

This time delay is small, but in principle can be measured by correlating the
electromagnetic waves at each detector, using the detectors as an extremely
large baseline for VLBI. As will be shown in Section 4, a couple factors
conspire so that for wide bandwidth measurements the time delays likely can be
measured with an uncertainty of $\sigma_{t}\sim\nu^{-1}$ for $\nu\gtrsim
3\,$GHz. The fractional distance uncertainty of a single measurement can then
be approximated as

$\displaystyle\frac{\sigma_{d}}{d}\sim\frac{\sigma_{t}}{\Delta
t_{d}}=0.17\%\left(\frac{x_{\perp}}{100~{}\text{AU}}\right)^{-2}\left(\frac{d}{100~{}\text{Mpc}}\right)\left(\frac{5~{}\text{GHz}}{\nu}\right)\left(\frac{\sigma_{t}}{\nu^{-1}}\right).$
(6)

Thus, the sensitivity scales with the square of the baseline, and baselines of
at least several AU are required to make cosmologically interesting distance
measurements at any frequency where VLBI has been performed. A system with a
baseline of 100 AU observing at 5 GHz could obtain sub-percent geometric
distance measurements to individual sources out to several hundred
megaparsecs.

The curvature wave front effect is related to the more astronomically familiar
parallax effect, as the extra delay from curvature can be thought of as
different baselines detecting the source at slightly different locations on
the sky. Indeed, if we have three equally spaced detectors each in a line with
a spacing of $x_{\perp}$, the parallax angle between the centers of the left
and right baseline is $x_{\perp}/d$ and each pair of adjacent antennas has an
angular resolution of $\sigma_{\theta}\sim c\,\sigma_{t}/x_{\perp}$. Therefore
$d$ can be measured to a fractional precision of
$\sigma_{\theta}/(x_{\perp}/d)\sim\sigma_{t}/\Delta t_{d}$, the same relation
we found from timing considerations. In the language of parallax, the reason
four detectors are needed is that each baseline localizes the position of a
source to a line with a girth of $\sigma_{\theta}$ (i.e. the angular position
is unconstrained perpendicular to the baseline). Thus, only when the three
detectors are in perfect alignment is this sufficient to measure the parallax,
but alignment is no longer required once a fourth detector is added.

##  3 FRBs as the best source candidate

Interferometric time delay measurements require the angular size of the source
to be smaller than (or at least comparable to) the angular resolution of the
baselines for the two antennas to receive identical (or nearly so) waveforms.
This means that the source must be smaller than
$\theta=\lambda/x_{\perp}=8\times 10^{-4}~{}({\rm
100AU}/x_{\perp})(\nu/5~{}\text{GHz})^{-1}~{}\mu$-arcsecond, where
$\lambda=c/\nu$ is the wavelength, or equivalently that the physical size of
the source must satisfy333Technically the $d$ that appears here is the angular
diameter distance, unlike in all other sections, where (in flat space) it is
the comoving light travel distance.

$\displaystyle\ell\lesssim\frac{\lambda d}{x_{\perp}}=1\times
10^{12}~{}\left(\frac{\rm
100~{}AU}{x_{\perp}}\right)~{}\left(\frac{\nu}{5\,{\rm
GHz}}\right)^{-1}~{}\left(\frac{d}{100~{}\text{Mpc}}\right)~{}\text{cm}.$ (7)

We can rewrite this equation in terms of the fractional distance and timing
errors:

$\displaystyle\ell$ $\displaystyle\lesssim$
$\displaystyle\frac{1}{2}(\sigma_{d}/d)(\nu\,\sigma_{t})^{-1}x_{\perp},$ (8)
$\displaystyle\lesssim$
$\displaystyle\sqrt{\frac{c\,d\;\sigma_{d}/d}{2\,\nu^{2}\,\sigma_{t}}}=10^{12.5}\left(\frac{d}{100\,\text{Mpc}}\right)^{1/2}\left(\frac{\sigma_{d}/d}{0.01}\right)^{1/2}\left(\frac{\sigma_{t}}{\nu^{-1}}\right)^{-1/2}\left(\frac{\nu}{5\,\text{GHz}}\right)^{-1/2}{\rm
cm},$ (9)

where we used that $\sigma_{d}/d=\sigma_{t}/\Delta t_{d}\sim
2\,c\,d\,\sigma_{t}/x_{\perp}^{2}$ and have grouped terms in a manner that
reflects the timing noise $\sigma_{t}$ is likely to be of the order of
$\nu^{-1}$ (§ 4). Equation 8 suggests that, for a timing precision of
$\sigma_{t}\sim\nu^{-1}$, the source needs to be $\gtrsim 100\times$ smaller
than the baseline separation for percent-level cosmology.

One potential source candidate in the sub-millimeter is the shadows of
supermassive black holes (Falcke et al., 2000; Event Horizon Telescope
Collaboration et al., 2019). Supermassive black hole shadows have sizes of
$\sim 3GM/c^{2}=4\times 10^{14}(M_{\rm BH}/10^{9}M_{\odot})~{}$cm. Thus, to be
unresolved in the sub-millimeter band at $\nu\sim 200~{}$GHz, a black hole
mass of $\lesssim 10^{6}({d}/{100\,\text{Mpc}})^{1/2}M_{\odot}$ is required
for 1% distance constraints if timing can be performed to
$\sigma_{t}\sim\nu^{-1}$. It is unlikely there are bright enough candidates at
these masses: This emission is relatively faint as a visible shadow requires
the accretion disk to be in a radiatively inefficient state, with the $6\times
10^{9}~{}M_{\odot}$ M87 black hole at just 16 Mpc having a flux density of
$\sim 0.5\,$Jy in the submillimeter and with this black hole being more
luminous than other similarly distant active galactic nuclei with compact
radio emission (Nyland et al., 2016). M87 is accreting at $\sim 10^{-5}$ its
Eddington limit, and a radiatively inefficient state may still be possible
even for $10^{-2}$, with the luminosity scaling quadratically with the mass
accretion rate (Narayan & McClintock, 2008), such that it is not impossible
that such a favorable class of black hole shadows exist.

Another source candidate is the faint radio afterglows of gamma ray bursts
(GRBs). The afterglow emission region has a size of $\ell\sim
c\,t\,\theta_{J}=3\times 10^{13}~{}(t/{\rm 1~{}day})(\theta_{J}/0.01)~{}$cm,
where measurements indicate that the smallest jet opening angles have
$\theta_{J}\sim 0.01$ (Piran, 2004). Therefore, even when observed a day after
the GRB, which is unusually early (Piran, 2004), GRB radio afterglows are
somewhat too large for this experiment.

Another possibility is to abandon the radio and do interferometry in the
optical, where distance estimates with precision $\sigma_{d}/d=0.01$ for a
source at $d=100\,$Mpc requires a baseline of _just_ $0.1~{}$AU if timing can
be performed to $\sigma_{t}=\nu^{-1}$. Now, looking past the impediment that
direct interferometry of the optical light from the two paths is likely
required, making achieving such baselines difficult to say the least, the most
obvious optical point source – quasars’ accretion disks – are still too large
for the compactness requirement of $\ell\lesssim 10^{10}$cm in the optical
band, with inferred sizes of thousands of AU (Morgan et al., 2010).

This brings us to FRBs, our best source candidate. Typical FRB durations of
milliseconds put an upper bound on the variability timescale, $\delta t_{\rm
var}$, which suggests that the underlying source is $\lesssim c\ \delta t_{\rm
var}$ or $\lesssim 1000$ km, $4-5$ orders of magnitude smaller than the size
requirement at gigahertz frequencies for baselines capable of precision
cosmology. Some FRBs even show variability on microsecond or even nanosecond
timescales (Nimmo et al., 2021, 2022). There is substantial evidence that FRB
emission originates from magnetars (which are highly magnetized neutron stars;
Petroff et al. 2019), including the detection of an FRB from a Galactic
magnetar (Bochenek et al., 2020; Andersen et al., 2020). In one class of
magnetar models, the FRB originates from near the $\sim 10~{}$km surface of
the neutron star (e.g. Kumar & Bošnjak, 2020). However, in the alternative
class of magnetar emission models, the FRB originates from a relativistic
shock further out (e.g. Metzger et al., 2019). In this model, the observed
emission region should be $\ell<(2\Gamma^{2}\,c\,\delta t_{\rm var})/\Gamma$,
i.e. larger by a factor of the Lorentz factor $\Gamma$ relative to $c\,\delta
t_{\rm var}$, but still smaller than we require for $\Gamma\lesssim 10^{4.5}$
if we take $\delta t_{\rm var}=1$ ms, which is well above what is allowed for
some FRBs, and the terms in parentheses in Equation 9 evaluate to unity
(furthermore the shock would be at an immense radius of $\gtrsim 10^{17}$cm).

Multi-path diffractive propagation from electron inhomogeneities in the FRB
host galaxy can make the effective image size larger.444This image broadening
also happens from plasma interactions within the Milky Way, which we discuss
in § 4.1. This effect is called “scattering.” The scattered image size is
related to the scattering timescale by $c\tau_{\rm sc}\sim\theta_{\rm
screen}^{2}d_{\rm screen}$, where $\theta_{\rm screen}$ is the angle from the
FRB to the illuminated part of the scattering screen at a distance $d_{\rm
screen}$. Then the effective size of the FRB emission region is
$\ell=\theta_{\rm screen}d_{\rm screen}=\sqrt{c\tau_{\rm sc}d_{\rm
screen}}=1\times 10^{11}(d_{\rm screen}/0.1\;{\rm pc})^{1/2}(\tau_{\rm
sc}(1~{}{\text{GHz}})/1\;{\rm ms})^{1/2}(\nu/5\,{\rm GHz})^{-2}$ cm, assuming
that the scattering time scales in the standard way as $\nu^{-4}$ (§ 4.1). A
$\tau_{\rm sc}(1\ \text{GHz})\sim 1$ ms scattering timescale contributed by
the host galaxy is detected in a significant fraction of FRBs (Thornton et
al., 2013; Qiu et al., 2020), although for some $\tau_{\rm sc}(1\ \text{GHz})$
is constrained to be $\lesssim 1\,\mu$s at $\nu=1~{}$GHz (Nimmo et al., 2021;
Cho et al., 2020). The choice $d_{\rm screen}=0.1\;$pc is motivated by the
radius of the magnetized nebula hundreds of years after the supernovae, a
timescale which explains the persistent radio emission observed around some
repeating FRBs (Margalit et al., 2018). Indeed Marcote et al. (2017)
constrained with VLBI the persistent radio emission to be $<0.7~{}$pc for FRB
121102. In conclusion, unless the millisecond scattering occurs at $d_{\rm
screen}\gtrsim 60(\nu/5\;{\rm GHz})^{3}$ pc , the scattered image of an FRB at
$d=100~{}$Mpc is likely sufficiently small.

##  4 Timing measurement accuracy

The proposed experiment correlates the electromagnetic waves at each detector,
using the detectors as an extremely large baseline for VLBI. Often for VLBI
astrometry, timing is done using the group delay, which is the rate of change
of phase over the band (not requiring the absolute phase). This leads to a
timing error of

$\displaystyle\sigma_{t,\rm group}^{\rm VLBI}$
$\displaystyle=\frac{1}{2\pi\Delta\nu_{\rm RMS}~{}\text{SNR}},$ (10)
$\displaystyle=0.016~{}\textrm{ns}~{}\left(\frac{\Delta\nu_{\rm
RMS}}{1~{}\textrm{GHz}}\right)^{-1}\left(\frac{\mathrm{SNR}}{10}\right)^{-1},$
(11)

where SNR is the signal-to-noise ratio at which correlations between two
detectors are detected and $\Delta\nu_{\rm
RMS}^{2}\equiv\text{SNR}^{-2}\\{\int d\nu\,\nu^{2}\,d({\rm
SNR}^{2})/d\nu-[\int d\nu\,\nu\,d({\rm SNR}^{2})/d\nu]^{2}\\}$ is the
effective RMS bandwidth squared (Rogers, 1970; Thompson et al., 2017).555If
correlating two instruments with different flux sensitivities, the SNR that
appears in Eqn. 11 is the geometric mean of the SNR of each instrument to
detect the flux. For uniform signal to noise over a continuous bandwidth of
$\Delta\nu$ then $\Delta\nu_{\rm RMS}=\Delta\nu/(2\sqrt{3})$. While the group
delay is less prone to systematics (namely clock phases and the Earth’s
atmosphere), using the absolute phase improves the timing noise over
$\sigma_{t,\rm group}^{\rm VLBI}$ by approximately the factor $\Delta\nu_{\rm
RMS}/\nu$. One gains from using the phase delay once the space-time detector
positions are known more precisely than the group delays, although sufficient
bandwidth is needed to discriminate the correct phase fringe
($\nu/\Delta\nu\ll{\rm SNR}$). See Appendix B for more discussion and we
return to this timing when we discuss dispersion in § 4.2.

Other considerations also affect the timing precision. Namely, inhomogeneities
in the intervening plasma that alter the path traveled by light (scattering;
§4.1) and ones that result in a relative difference in the average group
velocity between the detectors’ sightlines (differential dispersion; § 4.2).
Additionally, gravitational time delays turn out to be important for baselines
of $x\gtrsim 100\,$AU (§ 4.3).

###  4.1 Scattering

Scattering refers to the geometric time delay and smearing of the arrived
waveform from density inhomogeneities in the intervening plasma. The Milky
Way’s interstellar medium (ISM) can lead to different delays and smearing
along the sightline to each detector. Observations of pulsars and some FRBs
above the Galactic disk suggest a scattering time of

$\sigma_{t}^{\rm sc}\sim
30~{}\textrm{ns}~{}\left(\frac{\nu}{1~{}\textrm{GHz}}\right)^{-4}\sin(b)^{-2.5}$
(12)

for Galactic latitude ($b$) sightlines above the disk, with sightlines having
a factor of several scatter around this ‘average’ (Cordes & Lazio, 2002;
Cordes et al., 2016; Nimmo et al., 2021, 2022, the coefficient in Equation 12
is the NE2001 model value given in the first reference). The frequency scaling
of $\sigma_{t}^{\rm sc}$ is important as most measurements of scattering are
at smaller frequencies than are ideal for the proposed experiment. We adopt a
$\nu^{-4}$ scaling in Equation 12, which is used in our later forecasts. A
slightly steeper scaling of $\nu^{-4.4}$ is the expectation from Kolmogorov
turbulence (Blandford & Narayan, 1985). In models where the scattering owes to
the refractive lensing of a small number of current sheets (which have been
developed to explain pulsar scintillation data; Pen & Levin, 2014), the
average scaling would be $\nu^{-4}$ and with system-to-system scatter about
this scaling. Measurements are on-average consistent with $\nu^{-4}$ and show
significant system-to-system scatter (Löhmer et al., 2004; Krishnakumar et
al., 2019; Oswald et al., 2021).

For the baselines and frequencies we are considering, the structures that
contribute to the scattering at each antenna are approximately uncorrelated,
as this scattering time corresponds to light taking paths that differ by a
transverse length of

$\displaystyle R_{T}$ $\displaystyle=\sqrt{2\,L\,c\,\sigma_{t}^{\rm sc}}\sim
0.006~{}\text{AU}~{}\sin(b)^{-1.3}\left(\frac{\nu}{5~{}\text{GHz}}\right)^{-2}\left(\frac{L}{1~{}\text{kpc}}\right)^{1/2}.$
(13)

Since $R_{T}$ is much less than our $x\gg 1$ AU baseline lengths for relevant
frequencies, the scattering seen by each detector will be approximately
uncorrelated.

When the scattering is strong ($\sigma_{t}^{\rm sc}\gtrsim\nu^{-1}$), the
light travels over multiple paths with significant phase differences resulting
in the paths interfering at the antennas. This interference smears out the
electric field correlations, making them harder to detect (Narayan, 1992).
Thus, it becomes increasingly difficult with decreasing frequency to measure
the time delays between different detectors. When $\sigma_{t}^{\rm
sc}\lesssim\nu^{-1}$ the different paths add constructively and the scattering
becomes weak. Using Equation 12, this corresponds to $\nu\gtrsim
3\sin(b)^{-0.8}~{}$GHz. Weak scattering has the effect of adding an additional
delay to each sightline but not smearing out temporally the correlations. For
a given system, the frequency scaling of $\sigma_{t}^{\rm sc}$ should be the
same across the weak and strong limits when there are enough structures so
that the phase field is Gaussian random (Goodman & Narayan, 1989) and,
furthermore, weak refractive scattering always scales as $\nu^{-4}$.

In conclusion, Milky Way ISM scattering favors targeting $\nu\gtrsim 3\,$GHz,
as at lower frequencies scattering will exceed our $\nu^{-1}$ timing-noise
goal and additionally its interfering nature will make $E$-field correlations
more challenging to detect.

###  4.2 Differential dispersion

Dispersion in interstellar plasma will introduce a frequency-dependent time
delay owing to the wave front traveling with a group velocity that is less
than the speed of light. This delay is equal to
$\tau_{d}=\kappa\int_{0}^{d}n_{e}\,dx_{\parallel}$, where
$\kappa={e^{2}}/(2\pi\,m_{e}\,c\,\nu^{2})$. Here $n_{e}$ is the electron
density and this integral is performed along the entire line of sight to the
source. For observations at 5 GHz, and a typical
$\int_{0}^{d}n_{e}\,dx_{\parallel}$ (called the ‘dispersion measure’) in
current FRB samples of $500~{}\textrm{pc}~{}\textrm{cm}^{-3}$, the
corresponding time delay is 100 ms. For this experiment the overall $\tau_{d}$
does not affect timing, but differences in the $\tau_{d}$ between the
radiometer sightlines do. If large enough – which we show is likely –, these
delays need to be fit for and removed. (The higher-order effect from
deviations from straight-line paths is included in the scattering delay
considered in § 4.1.)

The largest contribution to $\tau_{d}$ differences to each detector is likely
turbulence in the ionized ISM. Appendix C considers this case for a power
spectrum parameterized as $P_{e}(k)=(2\pi)^{3}C_{n}^{2}\,k^{-11/3}$ and finds
that the contribution that looks like wave front curvature is

$\displaystyle\sigma_{t}^{\rm disp}\approx
90~{}\textrm{ns}\left(\frac{L}{0.1~{}\textrm{kpc}}\right)^{1/2}\left(\frac{x_{\perp}}{100~{}\textrm{AU}}\right)^{5/6}\left(\frac{\nu}{5~{}\textrm{GHz}}\right)^{-2}\left(\frac{C_{n}^{2}}{5\times
10^{-17}\textrm{cm}^{-20/3}}\right)^{1/2}.$ (14)

In the Solar neighborhood measurements find $C_{n}^{2}\sim 5\times
10^{-17}\textrm{cm}^{-20/3}$ (Armstrong et al., 1981; Draine, 2011). We have
chosen the $L$ so that when the parentheses evaluate to unity the scattering
measure SM$\equiv C_{n}^{2}L=5\times 10^{-18}\text{kpc~{}cm}^{-20/3}=1\times
10^{-4}~{}$kpc m-20/3 aligns with high Galactic latitude values in the NE2001
model (Cordes & Lazio, 2002). Models where the scattering does not owe to
turbulence but instead AU-scale current sheets (which better describe pulsar
scintillation data) would result in smaller $\sigma_{t}^{\rm disp}$ owing to
the bluer $P_{e}$ (Goldreich & Sridhar, 2006; Pen & Levin, 2014).

In addition, more local gaseous structures can contribute to the dispersive
delays. The differential electron column from the heliosphere at $R_{\rm
HS}\approx 100\,$AU can give rise to a time delay just like ISM density
fluctuations. Outside the heliosphere, the density rises by almost a factor of
one hundred to $n_{\rm IS}=0.1\,$cm-3 (Gurnett et al., 2013). The differences
in electron column from these positional differences result in timing
differences of

$\sigma_{t}^{\rm disp}\sim\kappa\frac{n_{\rm IS}x_{\perp}^{2}}{R_{\rm
HS}}=0.5~{}\text{ns}~{}~{}\left(\frac{\nu}{5~{}\text{GHz}}\right)^{-2}\,\left(\frac{x_{\perp}}{100~{}\text{AU}}\right)^{2}.$
(15)

While this component scales similarly with $x_{\perp}$ as $\Delta t_{d}$, it
is also small. Additionally, if a terrestrial antenna is used in the proposed
constellation, the Solar wind leads to several nanosecond dispersive delays at
$5\,$GHz (Border & Paik, 2009).

There are also timing differences from cosmological structures such as
circumgalactic clouds, but we find that these are negligible owing to their
large extents.

Pulsar timing arrays empirically constrain dispersion variations. Most pulsars
followed by the NANOGrav timing array show variations in their dispersion
measure of $\sim 100~{}{\rm ns}~{}({5~{}\text{GHz}}/\nu)^{2}$ over $\sim
10\,$yr of monitoring, which probes electron variations on lengths of $\sim
100~{}$AU (Jones et al., 2017). For most, the dispersion changes may be
dominated by the pulsar motion in its overdense vicinity, but the variation is
surprisingly close to our estimate from ISM turbulence.

Thus, the differential dispersive delays are likely to be much larger than the
$\nu^{-1}$ timing precision specification at the $\nu\lesssim 10~{}$GHz at
which FRBs have been detected. Fortunately, dispersive delays can be removed
by fitting simultaneously for them using their distinct $\nu^{-2}$ frequency
dependence. Fitting out differential dispersion requires a wide bandwidth to
not incur too large of a cost to the wave front timing sensitivity. In
Appendix B, we find that the timing noise with marginalization over dispersion
is given by $\sigma_{t,\rm group}^{\rm VLBI}/2$ in the case where the absolute
phase is used (and by $10\,\sigma_{t,\rm group}^{\rm
VLBI}\,[0.2\,\nu]/\Delta\nu$ in the less applicable case where only the group
delay is used), assuming a continuous bandwidth of $\Delta\nu$ and uniform
signal to noise over this bandwidth. This phase delay timing error increases
over the case without fitting for dispersion by a factor of
$\approx\nu/(2\,\Delta\nu_{\rm RMS})$. _Importantly, the error bars from
correcting the dispersive delays do not depend on the size of these delays._
This independence is well known from GNSS applications, where not correcting
for the relative dispersion from Earth’s atmosphere can result in $20~{}$m
errors.

###  4.3 Gravitational time delays

There will be stars, Solar System bodies, and even large-scale structure along
the line of sight to an FRB that will introduce different gravitational time
delays to each detector. For a mass at a distance $d_{M}$ for which the impact
parameter is much less than both
$\\{d_{M},|{\boldsymbol{d}}-{\boldsymbol{d}}_{M}|\\}$, the difference in the
Shapiro time delays between the detector at ${\boldsymbol{x}}$ and the one at
origin is given by (Shapiro, 1964):

$\displaystyle\Delta t_{\text{grav}}$
$\displaystyle=\frac{2GM}{c^{3}}\text{ln}\frac{(b+\delta_{b})^{2}}{b^{2}},$
(16)

where $M$ is the mass of the star, ${\boldsymbol{b}}$ is the impact parameter
for the first path (perpendicular to ${\boldsymbol{d}}$), and

$\delta_{b}=|{\boldsymbol{b}}+{\boldsymbol{x}}-({\boldsymbol{x}}\cdot\widehat{{\boldsymbol{d}}})\widehat{{\boldsymbol{d}}}|-b\approx\widehat{{\boldsymbol{b}}}\cdot{\boldsymbol{x}}+\frac{1}{2b}\left[x^{2}-({\boldsymbol{x}}\cdot\widehat{{\boldsymbol{d}}})^{2}\right]-\frac{1}{2b}\left(\widehat{{\boldsymbol{b}}}\cdot{\boldsymbol{x}}\right)^{2}+{\cal
O}\left(\frac{x^{3}}{b^{2}}\right)$ (17)

is the difference in the impact parameter for the second path, which this
expression relates to ${\boldsymbol{x}}$. (Recall that ${\boldsymbol{x}}$ is
our antenna separation vector.) Hats denote unit vectors, and vectors are the
same as indicated in Figure 2. The term
$-({\boldsymbol{x}}\cdot\widehat{{\boldsymbol{d}}})\widehat{{\boldsymbol{d}}}$
projects out the line of sight component of ${\boldsymbol{x}}$. Assuming that
the difference in impact parameter is small relative to the impact parameter
itself, we can expand Equation 16 in terms of $\delta_{b}/b$ to obtain

$\displaystyle\Delta t_{\text{grav}}$
$\displaystyle=\frac{2GM}{c^{3}}\left(\frac{2\delta_{b}}{b}-\frac{\delta_{b}^{2}}{b^{2}}\right)+\mathcal{O}\left(\frac{\delta_{b}^{3}}{b^{3}}\right),$
(18)
$\displaystyle=\frac{4GM}{c^{3}b^{2}}\left({\boldsymbol{b}}\cdot{\boldsymbol{x}}+\frac{1}{2}\left[x^{2}-({\boldsymbol{x}}\cdot\widehat{{\boldsymbol{d}}})^{2}\right]-\frac{\left({\boldsymbol{b}}\cdot{\boldsymbol{x}}\right)^{2}}{b^{2}}+{\cal
O}\left(\frac{x^{3}}{b}\right)\right),$ (19)
$\displaystyle=\frac{4GM}{c^{3}b^{2}}\left(b\,x_{\perp}\cos(\theta)-\frac{1}{2}x_{\perp}^{2}\cos(2\theta)+{\cal
O}\left(\frac{x^{3}}{b}\right)\right),$ (20)

where $\theta$ is the angle between ${\boldsymbol{x}}_{\perp}$ and
${\boldsymbol{b}}$. The last line used

$\left[x^{2}-({\boldsymbol{x}}\cdot\widehat{{\boldsymbol{d}}})^{2}\right]-2\left({\boldsymbol{x}}\cdot\widehat{{\boldsymbol{b}}}\right)^{2}=x_{\perp}^{2}-2\left({\boldsymbol{x}}_{\perp}\cdot\widehat{{\boldsymbol{b}}}\right)^{2}=-x_{\perp}^{2}\cos(2\theta),$

Equation 20 organizes the expansion into a dipole, quadrupole, etc.

The linear-in-$x_{\perp}$ dipolar term accounts for the majority of the
contribution to $\Delta t_{\text{grav}}$. However, the linear term is
identical to changing the direction of the FRB, which had resulted in
$-{\boldsymbol{x}}\cdot\widehat{{\boldsymbol{d}}}/c$ in $\Delta t_{d}$ in
Equation 3. Thus, the linear effect of the Shapiro time delay manifests as a
small shift in the apparent direction of the source so that it appears in the
direction
$(\widehat{{\boldsymbol{d}}}-\alpha\,\widehat{{\boldsymbol{b}}})/\sqrt{1+\alpha^{2}}$,
where $\alpha=4GM/(c^{2}b)$, which one might recognize as the deflection angle
from lensing. The effective $\alpha$ when accounting for the accumulation of
all gravitational deflections is dominated by the large-scale structure of the
Universe for sources at cosmological distances, resulting in arcminute-scale
deflections for sources at high redshifts – well known from studies of the
cosmic microwave background (Lewis & Challinor, 2006).666A worry is that the
full linear-in-$x$ delay when adding the Shapiro effect to the geometric delay
is
$(\widehat{{\boldsymbol{d}}}-\alpha\,\widehat{{\boldsymbol{b}}})\cdot{\boldsymbol{x}}/c$,
where $(\widehat{{\boldsymbol{d}}}-\alpha\,\widehat{{\boldsymbol{b}}})$ is not
a unit vector, suggesting we might need an additional timing constraint to
constrain the third component. The non-unit vector term to leading order in
$\alpha$ equals
$\alpha^{2}\widehat{{\boldsymbol{d}}}\cdot{\boldsymbol{x}}/(2c)$, which is
relatively large for cosmological values of the deflection angle $\alpha$.
However, the presence of this term is confusing as the linear Shapiro effect
acts to tilt the wave front, which should be describable with a unit vector.
The resolution is that the Shapiro effect does not account for the additional
path length of the tilted wave front. Accounting for this exactly cancels this
term.

The quadratic-in-$x_{\perp}$ quadrupolar Shapiro term is a source of noise for
our distance measurement, with a value given by

$\displaystyle\Delta t_{\text{grav}}^{\text{quad}}$
$\displaystyle=-0.025~{}\textrm{ns}\cos(2\theta)\left(\frac{x_{\perp}}{100~{}\textrm{AU}}\right)^{2}\left(\frac{M}{M_{\odot}}\right)\left(\frac{b}{1~{}\textrm{ly}}\right)^{-2},$
(21)
$\displaystyle=-0.023~{}\textrm{ns}\cos(2\theta)\left(\frac{x_{\perp}}{100~{}\textrm{AU}}\right)^{2}\left(\frac{M}{10^{13}M_{\odot}}\right)\left(\frac{b}{1~{}\textrm{Mpc}}\right)^{-2}.$
(22)

Let us concentrate first on the stellar case for which Equation 21 is
expressed. The stellar surface density from the Solar Circle out of the
Galactic disk is estimated to be $2\;M_{\odot}\,$Lyr-2 (McKee et al., 2015).
For the case of Poisson sampling of stellar locations from a spatially uniform
probability density, the quadratic Shapiro term will be dominated by the
closest stars to a sightline and so the parameters that yield $\Delta
t_{\text{grav}}^{\text{quad}}=-0.025\,\cos(2\theta)~{}$ns in Equation 21
should be in line with the typical Shapiro delay from stars in the Galactic
disk.777If there is a stellar structure that has a large quadrupole moment,
the collective contribution of many stars would be important. The most obvious
structure is the Galactic disk; however, the Disk mostly contributes a dipole
(a gradient) for high-latitude sightlines.

Regarding dark matter halos and large-scale structure, most sightlines out to
a gigaparsec will pass within the 300 kpc virial radius of a $\gtrsim
10^{12}M_{\odot}$ halo (McQuinn, 2014). Thus, this fact combined with Equation
22 suggests that the delay from dark matter halos is comparable to that of
Disk stars. However, a more detailed calculation in Appendix D shows that the
dark matter halo contribution is generally larger (with an RMS of $0.1\;$ns
for $d=100\;$Mpc and $x_{\perp}=100\;$AU) and increases roughly as the square
root of the distance to the FRB.

The $\propto x_{\perp}^{2}$ dependence of the quadratic Shapiro delay means
that, if this term is not removed, eventually increasing the baseline length
does not increase the precision to which $\Delta t_{d}$ can be measured. We
find in § 7 that the Shapiro delay can become the dominant noise source at
$x_{\perp}\gtrsim 100\,$AU. As the quadratic Shapiro term is quadrupolar (c.f.
Equation 22), it is completely specified by two angular moments such that two
additional detectors over the minimal four-detector constellation would enable
isolating and subtracting it.

Now lets turn to near-field Shapiro delays from nearby Solar System bodies.
For such bodies we cannot take the previous distant mass limit as $b\sim
d_{M}\ll d$, where $d_{M}$ is the distance from the body to a detector. We use
primed values for these distances to a second detector, where again
$b^{\prime}\sim d_{M}^{\prime}$. Thus, we require a more complete expression
for the Shapiro delay, given by

$\Delta
t_{\text{grav},\text{SS}}=\frac{2GM}{c^{3}}\left[\ln\left(\frac{d_{M}+\sqrt{d_{M}^{2}+b^{2}}}{{d^{\prime}}_{M}+\sqrt{{d_{M}^{\prime}}^{2}+{b^{\prime}}^{2}}}\right)-\frac{1}{2}\left(\frac{d_{M}}{\sqrt{d_{M}^{2}+b^{2}}}-\frac{{d^{\prime}}_{M}}{\sqrt{{d^{\prime}}_{M}^{2}+{b^{\prime}}^{2}}}\right)\right].$
(23)

In this nearby limit, the relative Shapiro time delay difference between two
baselines is of order $2GM/c^{3}$. In the extreme scenario in which all the
mass in the outer Solar System were closer to one baseline than to another,
the total effect would be of the order of $2G\,M/c^{3}=3\times
10^{-2}(M/M_{\earth})~{}$ns. The Kuiper Belt is estimated to have a mass of
$10^{-2}-0.1\,M_{\earth}$ and a similar mass is for known trans-Neptunian
objects (Pitjeva & Pitjev, 2018). Thus, this amount of mass does not lead to
problematic delays, and really the systematic will be smaller than the above
estimate, likely reduced by the square root of the effective number of systems
that contribute (which for the $\sim 100$ trans-Neptunian objects would be a
factor of $\sim 10$ suppression).

The hypothesized “Planet 9,” with a mass of $\sim 10\,M_{\earth}$ and an orbit
of $\sim 500~{}$AU (Batygin et al., 2019), would lead to timing noise at the
$0.1~{}$ns level and could be a concern. However, its mass would likely be
measured and corrected by our proposed instrument, both by how its gravity
deflects the orbits of the antennas (Witten, 2020, and § 8) and by its effect
on their onboard clocks (§ 4).

##  5 FRB Rate

The clearest path is to directly target the known population of repeating
FRBs, for which about fifty are currently known (Fonseca et al., 2020; Petroff
et al., 2022; The CHIME/FRB Collaboration et al., 2023, which has doubled
since this article was released as a pre-print!). The current sample of
repeating FRBs with a known galactic host include FRB 20121102A (the original
repeater) at a comoving light-travel distance of $d=800$ Mpc, FRB 20180916B at
$d=150$ Mpc, FRB 20180301A and FRB 20190520B are somewhat further away with
$d\approx 1000$ Mpc, FRB 20181030A is most likely at $20~{}$Mpc, and FRB
20201124A at $400$ Mpc (Petroff et al., 2022; Bhardwaj et al., 2021). Bright
repeating FRBs are at distances that are amenable to the proposed experiment,
with the number of confirmed repeating FRBs constituting 2.6% of all FRB
sources discovered by the CHIME collaboration (The CHIME/FRB Collaboration et
al., 2023). Most of these are inferred to repeat on a timescale of a few hours
(CHIME/FRB Collaboration et al., 2019; The CHIME/FRB Collaboration et al.,
2023), and they are known to show spates of intense activity – FRB20201124A
was clocked at several registered bursts per hour (Main et al., 2021)! As
shown below, the repeating bursts generally have flux densities at $\sim
1\,$GHz that our fiducial constellation parameters of four satellites each
equipped with 10 m dishes would be sensitive to if they have similar flux
densities at the targeted frequency. Furthermore, if the FRB is detected on
Earth by a more sensitive instrument, this voltage time series could be used
to effectively enhance the SNR as the geometric mean of the two instruments.

While we are hopeful that a sufficient sample of repeating FRBs will be found
for the proposed experiment, it could also detect FRBs that occur in its field
of view. We estimate here the FRB detection rate assuming that the array is
not assisted by a more sensitive terrestrial telescope. The minimum specific
fluence the proposed constellation is sensitive to is (e.g. Burke & Graham-
Smith, 2009):

$\displaystyle F_{\nu}^{\rm min}$ $\displaystyle=$
$\displaystyle\frac{8kT_{\textrm{sys}}\tau}{{\pi D^{2}}N_{\rm
sat}\sqrt{\Delta\nu\tau}}\,\text{SNR}_{\rm tot},$ (24) $\displaystyle=$
$\displaystyle
1.4\,\textrm{Jy~{}ms}\left(\frac{D}{10~{}\textrm{m}}\right)^{-2}\left(\frac{N_{\rm
sat}}{4}\right)^{-1}\left(\frac{\Delta\nu}{1~{}\textrm{GHz}}\right)^{-1/2}\left(\frac{\tau}{1~{}\textrm{ms}}\right)^{1/2}\left(\frac{T_{\text{sys}}}{20~{}\text{K}}\right)\left(\frac{\text{SNR}_{\rm
tot}}{8}\right),$ (25)

where $D$ is the diameter of each dish, $N_{\rm sat}$ is the number of
satellites, $\Delta\nu$ is the bandwidth, $\tau$ is the effective duration of
the FRB, $F_{\nu}$ is the specific fluence, $T_{\textrm{sys}}$ is the system
temperature of the receiver, and $\text{SNR}_{\rm tot}$ the total signal to
noise at which the array detects the FRB. At gigahertz frequencies,
$T_{\textrm{sys}}\approx 20$ K is commonly achieved (Burke & Graham-Smith,
2009). Such a radio telescope would have a beam with a solid angle of
$\Omega_{\rm beam}={4c^{2}}/({\pi D^{2}\nu^{2}})=5\times
10^{-5}~{}\textrm{sr}\left({D}/{10~{}\textrm{m}}\right)^{-2}\left({\nu}/{5~{}\textrm{GHz}}\right)^{-2}$,
although, at the expense of large data rates and computation that may not be
possible in space, modern radio dishes can be phased to generate hundreds
beams, $N_{\rm beam}$, for a total solid angle of $\Omega_{\rm tot}=N_{\rm
beam}\Omega_{\rm beam}$.

The all-sky rate of FRBs can be modeled as a power law where the rate above
some fluence limit is given by $R(>F_{\nu})\propto F_{\nu}^{-\gamma}$. The
CHIME survey has inferred $800$ FRBs occur per day above 5 $\rm Jy\cdot ms$
with $\gamma=1.4\pm 0.2$ (Amiri et al., 2021), near the Euclidean-space
scaling of $\gamma=3/2$. Adopting the Euclidean scaling, the rate of
detections for our antennas is

$\displaystyle R$
$\displaystyle=800~{}\textrm{day}^{-1}~{}K_{\nu}^{3/2}~{}\left(\frac{\Omega_{\rm
tot}}{4\pi}\right)~{}\left(\frac{F_{\nu}^{\rm
min}}{5~{}\textrm{Jy}\cdot\text{ms}}\right)^{-3/2},$ (26)
$\displaystyle=2.0~{}\textrm{day}^{-1}~{}~{}K_{\nu}^{3/2}\left(\frac{N_{\rm
beam}}{100}\right)\left(\frac{D}{10~{}\textrm{m}}\right)\left(\frac{N_{\rm
sat}}{4}\right)^{3/2}\left(\frac{\nu}{5~{}\textrm{GHz}}\right)^{-2}\left(\frac{\Delta\nu}{1~{}\textrm{GHz}}\frac{1~{}\textrm{ms}}{\tau}\right)^{3/4}\left(\frac{T_{\text{sys}}}{20~{}\text{K}}\frac{\text{SNR}_{\rm
tot}}{8}\right)^{-3/2},$ (27)

where $K_{\nu}$ is a bolometric correction factor that corrects on-average the
specific fluence for the observation as $\nu$ relative to the CHIME
$400-800~{}$MHz band. Observations of the first repeater FRB 20121102 at 8 GHz
show similar flux densities to those at $1\,$GHz (Gajjar et al., 2018).
Bethapudi et al. (2022) found a scaling of $F_{\nu}\propto\nu^{-1}$ for the
repeater FRB 20180916B over $4-8~{}$GHz (but with fewer repetitions at these
high frequencies relative to at $\sim 1\,$GHz) and a similar scaling is seen
up to $6\,$GHz for the repeater FRB 20190520B (Anna-Thomas et al., 2022). The
scaling $F_{\nu}\propto\nu^{-1.5\pm 0.3}$ has been measured between $1.1$ and
$1.4~{}$GHz for the average spectrum of a large number bursts (Macquart et
al., 2019). Even with the strong $-1.5$ scaling such that $K_{\nu}=0.1$ for a
$\nu=5\,$GHz observation, a rate of $R\approx 20\;$yr-1 is anticipated for the
fiducial specifications for which the parentheses in Equation 27 evaluate to
unity. This suggests that a non-negligible rate is not infeasible for $D>10$m
if $N_{\rm beam}$ that have been achieved on the ground are possible, despite
the relatively high frequencies of $\nu\gtrsim 3\,$GHz that we find are
required owing to Milky Way scattering.

The design also does not have to be radio dishes. Each satellite could instead
be a plane of dipoles or small dishes. If the effective collecting area is
comparable to $\pi D^{2}/4$, this would enable the same sensitivity as our
dish but allow up to $\Omega_{\rm tot}\sim 4\pi$ and a huge number of FRBs per
day! The additional cost to this approach is the computation required to phase
all the dipoles to point simultaneously in hundreds or thousands of
directions. This would require onboard computation that is well beyond that on
modern spacecraft.

If an individual satellite triggers on a burst, then it only needs to save the
electric field time series over the $\sim 1\;$ms duration of the FRB (or
instead the $\sim 0.1\,(5\,\text{GHz}/\nu)^{2}$ s duration if the dispersive
sweep is not corrected). However, if the burst is not flagged by the
satellite, the satellite either needs to record the electric field time series
for longer than the time it takes to communicate with Earth-based antennas,
which would signal that an FRB had occurred. This approach requires the
satellite to store at least $N_{\text{beams}}r/2\times
2\Delta\nu=10^{13}\,N_{\text{beams}}\,(r/10\,\text{AU})(\Delta\nu/1\,\text{GHz})$
numbers, each stored with at least one bit. (Even one bit is common in VLBI,
but comes with some signal loss; Thompson et al. 2017.) While this storage
demand is likely prohibitively high for $N_{\text{beams}}\gg 1$, if the signal
is flagged for a SNR threshold on a single antennas, then the storage cost can
be greatly reduced by a factor of $\sim\exp(-\text{SNR}^{2}/2)$ until all
storage is actual FRB signal. The $5~{}$GHz electric field of a millisecond-
long FRB requires relaying back just $\sim 10^{7}$ numbers for correlation.

##  6 Positional and temporal calibration

###  6.1 Sources of acceleration and calibration timescales

The outer Solar System is a lower acceleration environment than the medium-
Earth orbit of most global positioning satellites. Even at medium-Earth orbit,
global navigation satellites positions (ephemerides) are predicted to meter
accuracy one day in advance, allowing updated ephemeris information to be
uploaded to GNSS satellites only every several hours (Misra & Enge, 2012). A
primary factor limiting the precision of GNSS satellite ephemerides is various
stochastic accelerations, including variations in the Sun’s irradiance.
Stochastic acceleration sources also set the timescale that the satellite
ephemerides in the proposed experiment need to be calibrated, which in turn
inform the types of calibration systems that would be required. In what
follows, we estimate the magnitude of different sources of stochastic
accelerations for satellites in the outer Solar System. Sources of
gravitational acceleration are the most concerning, as an active system that
adjusts in response to an accelerometer the satellite orbit can potentially
correct for non-gravitational accelerations.

gravity from asteroids

Accelerations from the passages of asteroids that are close enough so they
occur over a much shorter timescale than their orbital time are the main
concern for calibration. Let us consider the requirements to have a sufficient
number of close passages for calibration to be necessarily on a timescale of
$t_{\text{calib}}$. The shorter $t_{\text{calib}}$, the more challenging for
calibration. In order to require calibration every $t_{\text{calib}}$,
sufficiently massive asteroids within a distance $v\,t_{\text{calib}}$ must be
commonplace, where $v$ is the satellite–asteroid relative velocity. For that
asteroid to displace the satellite by $1~{}$cm over a time $t_{\text{calib}}$,
it must have a mass satisfying $m>m_{\rm cr}\equiv v^{2}\times
1\,\text{\,cm}/G$, where $G$ is Newton’s constant or, assuming a mass density
of 2 g cm-3 and a spherical body, a radius of at least
$3\,(v/1\,\text{km~{}s}^{-1})^{2/3}~{}$km. Here $v\sim 1\,\text{km~{}s}^{-1}$
is motivated as, if our satellites are in orbit, we expect passage at a
fraction of the $\sim 5~{}$km s-1 Keplerian velocity at tens of AU radii owing
to the moderate eccentricities of asteroids and possibly of our satellites.
However, our favored configuration would be unbound satellites that are
drifting out at $10-100\,$km s-1, in which case gravitational accelerations
from asteroids would be even much smaller and due to larger masses. For
asteroid deflections to be commonplace requires a number density of at least
$n_{\rm KB}=(vt_{\text{calib}})^{-3}$ or a total number of $N_{\rm KB}\approx
4\pi/3\,r_{\rm KB}^{3}\sin(\theta_{\rm KB})n_{\rm KB}=4\times 10^{12}(r_{\rm
KB}/40\,\text{AU})^{3}(v/1\,\text{km~{}s}^{-1})^{-3}(t_{\text{calib}}/1\,\text{week})^{-3}\sin(\theta_{\rm
KB})$ asteroids above $m_{\rm cr}$, where $\theta_{\rm KB}$ describes the
asteroids’ angular extent above and below the ecliptic. For $\theta_{\rm
KB}=10^{\circ}$ and $r_{\rm KB}=40\,$AU characteristic of the Kuiper belt and
an orbital relative velocity of $v=1\,$km s-1, an anomalous acceleration that
results in a centimeter displacement over a week requires a number density of
kilometer-sized asteroids that is 4-5 orders of magnitude higher than in
models that match observations (Kenyon & Bromley, 2004; Schlichting et al.,
2009). (Optical observations probe asteroids with radii of $\gtrsim 10~{}$km,
with X-ray diffraction constraining the abundance at $\approx 200$ m.)
Similarly, the total mass we require is $N_{\rm KB}m_{\rm
cr}>100\,M_{\earth}\;(r_{\rm
KB}/40~{}\text{AU})^{3}(v/1\,\text{km~{}s}^{-1})^{-1}(t_{\text{calib}}/1\,\text{week})^{-3}~{}\sin(\theta_{\rm
KB})$, much larger than the $\sim 0.01-0.1~{}M_{\odot}$ estimated mass for the
Kuiper belt (Bernstein et al., 2004). If we instead calibrate to the $N_{\rm
KB}\sim 10^{8-9}$ asteroids with radii of $>3\,$km found in models that are
matched to Kuiper belt constraints (Kenyon & Bromley, 2004; Schlichting et
al., 2009), then following the above logic, this indicates a calibration time
of $t_{\text{calib}}\approx 10~{}$weeks. Thus, we estimate that the
calibration time for a satellite on a bound orbit _within_ the Kuiper belt is
$\sim 10~{}$weeks, and this time should be longer if the satellite resides
elsewhere or is unbound.

gaseous drag

The drag force on a surface of effective area $A_{\rm eff}$ traveling through
the Solar wind with densities of $n=0.05\,(r/10\,\text{AU})^{-2}\;$cm-3 and
particle velocities of approximately $v=500$ km s-1 (Villanueva, 1994) can be
computed as $F_{\rm drag}=1.4\,m_{p}\,n\,v^{2}\,A_{\rm eff}$. Drag results in
a 1 cm displacement after a time of $t_{\rm drag}=\sqrt{2M_{\rm sat}\times
1\,\text{cm}/F_{\rm drag}}=9~{}(A_{\rm eff}/10\text{m}^{2})^{-1/2}(M_{\rm
sat}/10^{3}\,{\rm kg})^{1/2}(r/30\,\text{AU})$ days. The Voyager spacecrafts
show factor of $\sim 2$ variations in the density of the Solar wind on the 27
day Solar rotation period (Villanueva, 1994; Richardson et al., 2003), which
would result in a somewhat longer time for the more relevant stochastic drag
relative to our homogeneous estimate. Neutrals from interplanetary space
penetrate the Solar System with the local interstellar abundance of
$0.1\,$cm-3 at a velocity of $30\,$km s-1 (Holzer, 1977). Their drag results
in a 1 cm displacement over $t_{\rm drag}\approx 40$ days, becoming the
dominant drag at $r\gtrsim 100\,$AU.

Lorentz force on a charged spacecraft

The spacecraft will inevitably build up a charge, especially since the radio
dish is a large conducting element. The maximum possible charge is roughly set
by that which is sufficient to repel Solar wind protons from striking and
further charging the spacecraft, namely $m_{p}v^{2}=Z_{\rm max}\,e^{2}/R$,
where $v$ is the velocity of intersecting particles – the $500\,$km s-1
velocity of the Solar wind – and $R$ is the effective extent of the satellite.
The Lorentz force, $Z_{\rm max}\,e\,(v_{\rm sat}/c)\,B$, acting on the
satellite traveling through an interplanetary magnetic field with
$B=1\,\mu\,G$ (Villanueva, 1994, and with factor of $\sim 2$ monthly
variations) results in a displacement of $1\,$cm after
$14\,(R/10\,\text{m})^{-1/2}(M_{\rm sat}/10^{3}\,{\rm kg})^{1/2}~{}(v_{\rm
sat}/50\,\text{km~{}s}^{-1})^{-1/2}(B/1\,\mu G)^{-1/2}\,$days. Spacecraft
charging would be less important outside of the heliosphere owing to the much
smaller velocities of the incident plasma.

solar radiation pressure

Solar radiation pressure variations are a primary reason for the meter
ephemeris errors on day timescales in global positioning satellites (Blewitt
et al., 1988). Even though the radiation pressure is down by a factor
$10^{-3}(r/30\,\text{AU})^{-2}$ in the outer Solar System relative to our
terrestrial environment, the centimeter ephemeris accuracy requirement is two
orders of magnitude more stringent. During active periods, the Sun shows 0.2%
peak-to-peak variations on the timescale of its 27 day rotation period
(Fröhlich & Lean, 2004). If we parameterize the correlation function of the
variations as $|\xi(t)-\xi(0)|=10^{-3}\,(t/13.5\,\text{days})^{1/2}$
–motivated by the scaling of the power spectrum of these fluctuations on
$1-10~{}$day timescales (Fröhlich & Lean, 2004) –, we find that fluctuations
in the solar irradiance will result in an acceleration of
$\sim|\xi(t)-\xi(0)|\,F_{\odot}A_{\rm eff}/M_{\rm sat}$ and, integrating this
acceleration with respect to time, a $1$ cm displacement of the satellite
every $\sim 11~{}(A_{\rm eff}/10~{}\text{m}^{2})^{-0.4}(M_{\rm
sat}/10^{3}~{}\text{kg})^{0.4}(r/30\,\text{AU})^{0.8}~{}\text{days}$, where
$F_{\odot}$ is the momentum flux of the Sun and $A_{\rm eff}$ is the effective
surface area of the satellite. Understanding thermal re-emissions of the
absorbed light is also important.

dust collisions

Collisions with dust particles will accelerate the spacecraft. The Ulysses and
Voyager spacecrafts found that outside of $5\,$AU the population of $\sim
0.1\,\mu$m dust was dominated by an interstellar population with impact speeds
of $v_{0.1\mu}=30\,$km s-1 and a resulting flux of $\approx
10^{-8}$cm${}^{-2}\,$s-1 (Gruen et al., 1994; Gurnett et al., 2005). Such
fluxes correspond to $N_{\rm coll}\approx 100\,(A_{\rm eff}/10\,\text{m}^{2})$
dust particles striking an effective area of $A_{\rm eff}$ per day, assuming
the spacecraft velocity is $\lesssim v_{0.1\mu}$. The stochastic (e.g.
$1/\sqrt{N_{\rm coll}}$) component of the acceleration is so small that it
takes hundreds of years to result in a 1 cm displacement for the Ulysses-
inferred average dust mass of $3\times 10^{-14}\,$g and a satellite mass of
$M_{\rm sat}=10^{3}\,$kg. More important are the larger dust grains seen in
zodiacal light. While outside of several AU the reflectance of these particles
contributes negligibly to this diffuse emission and hence the distribution of
such tens-of-micron-sized particles is unconstrained, within this radius it is
found that an order unity fraction of zodiacal particles is relatively
isotropic with density scaling radially as $\propto r^{-1.3}$ (Gruen et al.,
1985). For the following, we assume that this radial scaling still holds well
beyond several AU. If a $30\mu$m grain with mass $8\times 10^{-7}$g is
conservatively assumed with an abundance normalized to match the zodiacal
background ($2\times 10^{-17}$cm-3 at $5\,$AU; Gruen et al., 1994), one
particle would strike the satellite every $1.4\,(A_{\rm
eff}/10\,\text{m}^{2})^{-1}(r/10\,\text{AU})^{1.3}(v_{\rm sat}/100\,\text{km
s}^{-1})^{-1}$ days. A single particle would result in an acceleration that
leads to a 1 cm displacement after $1.4\,(M_{\rm sat}/10^{3}\,{\rm kg})(v_{\rm
sat}/100\,\text{km s}^{-1})^{-1}$ days. These estimates for the timescale for
zodiacal dust are somewhat smaller than the other acceleration timescales
considered above, but we have evaluated with very conservative values (a high
$v_{\rm sat}$, a small $r$ and a large mass per particle888The number density
of dust grains per log mass at an $1\,$AU is inferred to scale with particle
mass as $m^{-0.5}$ (Gruen et al., 1985). Our estimate assuming all particles
are in very large $30\,\mu$m grains overestimates the acceleration.). Thus, we
suspect that dust collisions again require calibration on a week or longer
timescales.

time dilation from Solar System masses

The mass distribution in the Solar System also affects onboard clocks. Clocks
run a factor of $(1-3r_{M}/[2r])$ slower when in orbit at distance $r$ around
an enclosed (spherically symmetric) mass of $M$ relative to the global
Schwarzschild time, where $r_{M}\equiv GM/c^{2}$ is the gravitational radius,
and the time dilation is similar for unbound trajectories. For a bound orbit,
a tenth of an Earth mass uncertainty in the enclosed mass, near the upper
limit on the mass in the Kuiper belt, would would result in a clock drift of
$\Delta t_{\text{clock}}=0.03~{}(t/1\,\text{week})(100\,\text{AU}/r)\,$ns
after a time $t$. Any uncertainty in the enclosed mass would be easy to
calibrate as the timescale for this drift is long. Consider instead a passing
outer Solar System object of mass $M$ and distance $d_{M}$ – such as some
Kuiper belt object. Its mass would contribute a clock drift rate of $\sim
r_{M}/d_{M}$. Using that the cumulative mass distribution of Kuiper belt
objects above mass $M$ is found to be $N(M)\sim M^{-2}$ (e.g. Schlichting et
al., 2009), we expect an object within a logarithmic mass interval in $M$ to
come within a distance of $d_{M}=N(M)^{-1/3}=M^{2/3}$. Thus,
$M/d_{M}=M^{1/3}$, suggesting that the largest bodies create the largest
timing errors. A worst case scenario might be an Earth-massed object at a
distance $\ell_{\rm sep}=1\,$AU from one detector, which would result in a
detectably large drift of $\sim 3$ ns day-1. Of course, this long-timescale
drift (or even ones that are hundreds of times smaller) should be detected in
calibration, and, hence, the $M/\ell_{\rm sep}$ of the object measured! Clock
drifts on week or month timescales from the gravity of (small) asteroids that
make close enough approaches are negligible.

In conclusion, variations in Solar irradiance, dust collisions, gaseous drag,
and possibly charging require calibration on $\gtrsim 10~{}$day timescales for
motivated specifications for the proposed satellite constellation.
Gravitational accelerations that result in a 1 cm displacement occur over
longer timescales. Gravitational clock drifts owing to passing bodies that
occur over weeks or months should be negligible.

###  6.2 Calibration methods

Measuring time delays to within (5 GHz)-1 = 0.2 ns implies that the relative
detector positions must be modeled to six centimeters. We suggest two possible
strategies for achieving such calibration. First, our favored calibration
strategy is direct trilateration between the satellites themselves. Our second
calibration method would combine satellite–Earth communications with
calibration off of astrophysical sources.

We first consider a calibration strategy that relies on GNSS-like
trilaterations. On Earth, these techniques regularly provide meter accuracy,
if not centimeter to millimeter with longer integrations that use the carrier
phase: Generally four satellites are needed for geolocation, but, if the user
possesses an accurate clock, only three are required. In this calibration
mode, the proposed experiment would be a much enlarged version of GNSS, where
members of the constellation would relay signals to measure each other’s
distances. Modulo signal to noise effects, enlargement itself does not affect
the distance accuracy.

First, let us consider the power requirements. The proposed experiment equips
each satellite with a radio dish, which is fortunate as this is also necessary
for the dishes themselves to broadcast strong enough signals for ranging.
Ranging works by modulating the carrier wave at frequency $\nu_{\rm car}$ with
some code with frequencies of $\sim\delta\nu$, which when multiplied by the
carrier approximately sets the bandwidth. For global positioning satellites,
$\delta\nu\sim 1\,$MHz is chosen to reduce computation and radio frequency
interference. The receiver then uses a matched filter with different temporal
delays to find the arrival time. The error in the ranging delay from this
‘delay-lock-loop’ procedure is (e.g. Misra & Enge 2012, see their section
10.5)

$\displaystyle\sigma_{t}^{\text{ranging}}$ $\displaystyle=$ $\displaystyle
T_{c}\sqrt{\frac{\epsilon\,k_{b}\,T_{\rm sys}}{4\,t\,P_{\rm rec}}},$
$\displaystyle=$
$\displaystyle{0.04\;}\text{ns\;}\left(\frac{10\,\textrm{MHz}}{T_{c}^{-1}}\right)\left(\frac{10\,\textrm{m}}{D}\right)^{2}\left(\frac{5\,\text{GHz}}{\nu}\right)\left(\frac{x}{100\,\rm
AU}\right)\sqrt{\left(\frac{\epsilon}{0.1}\right)\left(\frac{1\,\textrm{min}}{t}\right)\left(\frac{100\,\text{Watt}}{P_{\text{ant}}}\right)\left(\frac{T_{\text{sys}}}{20\,\text{K}}\right)},$

where we have used that the received power can be written as $P_{\rm rec}=\pi
D^{2}P_{\rm ant}/(4\Omega_{\rm beam}x^{2})$, where $P_{\rm ant}$ is the power
broadcast by the other antennas, and that the noise power is white with
$P_{N}=Bk_{B}T_{\rm sys}$, where $B$ is the bandwidth. Here $t$ is the
integration time, and $T_{c}$ is the duration of code chips such that
$T_{c}\sim 1/(2\delta\nu)$. (The code for GNSS applications are ‘chips’ of
zeros and ones with duration $T_{c}$. We are borrowing this approach.) The
parameter $\epsilon$ is the correlator spacing for the delay lock loop – the
circuit that attempts to phase up the satellites matched code with the
received signal. GNSS applications have to filter for radio frequency
interference and this generally restricts $\epsilon>0.1$ (Misra & Enge, 2012),
but it is likely in our case that smaller $\epsilon$ may be achievable with
the absolute minimum being $\epsilon\approx(BT_{c})^{-1}$ (Braasch & van
Dierendonck, 1999).999One can approximately derive Equation 6.2 from
optimizing the location of a matched filter. In ranging applications, the
correlation function of matched code with the signal offset by a time $\tau$
is chosen so that it is the triangle function scaling as $(1-\tau/T_{c})^{2}$,
as happens for a random sequence of chips. This results in a likelihood
function ${\cal L}$ of $-2\log{\cal L}=P_{\rm
rec}(1-\tau/T_{c})^{2}/(k_{b}T_{\rm sys}B)\times N$, where $N=2Bt$ is the
number of temporal samples and we have assumed one polarization. The standard
deviation which $\tau$ can be measured is then
$\sigma_{t}^{\text{ranging}}=(-\partial_{\tau}^{2}\log{\cal L})^{-1/2}$, which
yields the result to the factor of $\sqrt{\epsilon}$ or so. We suspect one
reason for the $\sqrt{\epsilon}$ improvement over the ‘ideal’ matched filter
is that the delay-lock-loop circuit has access to all lags and not the just a
grid of temporal samples that this matched filter estimate assumes. The above
is for ranging using the low-frequency code. Once the satellite is localized
to better than $c/\nu_{\rm car}$ so that integer wavelength ambiguities are
resolved, the carrier wave can then be used to measure the distance. The
carrier provides a precision that is even much higher and likely limited by
other considerations than SNR (Zhang et al., 2020).

However, it may also be possible to use a network of sources to calibrate the
distances rather than direct trilateration using our satellite network. Since
FRBs are likely the only extragalactic radio object that appear point-like at
Solar System-scale baselines (§ 3), they are also likely the only
extragalactic source that is useful for calibration. FRBs known to be more
distant than the distance reach of the array can be used for calibration (but
also FRBs with well-characterized distances). Each distant FRB provides
$N_{D}-3$ constraints after fitting for the source’s angle on the sky and the
FRB event time, where $N_{D}$ is the number of detectors. Repeating FRBs are
even more valuable, with each repetition providing $N_{D}-1$ additional
constraints once the relative motion of the repeating FRB is determined. The
goal of calibration is to constrain the $4N_{D}$ satellite space-time
coordinates. In the case $N_{D}=4$ (6), to calibrate the network requires $16$
($8$) one-off FRBs over a calibration time or, instead, 6 (5) repeating FRBs.
The calculations in § 5 suggest that it may be challenging to detects 5-16
FRBs within the $\sim 1~{}$week calibration time needed if non-gravitational
sources of acceleration cannot be removed. If so, a system that relies
predominantly on distant FRBs for calibration would need a precise
accelerometer to correct for non-gravitational accelerations.101010FRBs are
known to be highly polarized (Petroff et al., 2022), which potentially
provides an additional lever for calibration. The polarization should be the
same for all the detectors as the differential Faraday rotation on Solar
System-scale baselines is likely negligible; Faraday rotation is dominated by
large-scale coherent magnetic fields.

Similarly, pulsars may be the only common point-like Galactic source. See § 8,
as Galactic pulsars may not be sufficiently point-like, especially for the
longer baselines we consider. However, calibration off of pulsars would be
limited to the small fraction of pulsars at high Galactic latitudes to avoid
ISM scattering.

The clocks on the satellites in our constellation must be kept synchronized.
The clock requirements for centimeter space-time precision and weekly
calibration are somewhat more stringent than those for the atomic clocks on
GPS satellites, which err at several nanoseconds a day rather than the few
tenths of a nanosecond per week that would be required to not be the dominate
error in the ephemerides. Such clock specifications are somewhat more
stringent than the $4~{}$ns per $23$ days accuracy achieved by the Deep Space
Atomic Clock mission, a miniature atomic clock that was launched in 2019 to
demonstrate improved space-clock technology for future NASA missions as well
as for next-generation GPS (Burt et al., 2021). The best atomic clocks in the
world exceed the performance of the Deep Space Atomic Clock by many orders of
magnitude (Ludlow et al., 2015). Less accurate clocks than the tenths of
nanosecond per week specification entails more regular calibration or $\gtrsim
1\,$cm ephemerides errors.

##  7 Distance measurement accuracy

The previous discussion motivates that the distance-dependent time delays
$\Delta t_{d}$ may be measurable with a Solar System-scale interferometer with
an uncertainty that can approach $\sim\nu^{-1}$ or even better. Figure 3 shows
the various time delays calculated in § 4 versus projected baseline length
$x_{\perp}$ for $\nu=4\,$GHz (top panel) and $\nu=8\,$GHz (bottom panel). The
solid lines are the cosmological signal of interest that owes to wave front
curvature, $\Delta t_{d}$, for $d=100\,$Mpc (blue) and $d=1000\,$Mpc (red).
The other lines are estimates for different systematic delays: the scattering
time (taken to be $\sigma_{t}^{\rm
sc}=50\,\text{ns}\,[\nu/1\,\text{GHz}]^{-4}$, $2\times$ larger than the mean
NE2001 model at $b=90^{\circ}$; Equation 12), the Shapiro time delay (assuming
the cosmological delay is dominant; Equation D4), the VLBI group delay timing
error $\sigma_{t,\rm group}^{\rm VLBI}$ (assuming SNR$=5$ for the baseline
with $d{\rm SNR}/d\nu$ uniform over a band of $\Delta\nu=0.2\,\nu$; Equation
11), and the differential dispersion from the Galaxy (Equation 14 assuming
SM$\equiv C_{n}^{2}L=10^{-4}\;\textrm{kpc~{}m}^{-20/3}$). While the
differential dispersion is even larger than our signal $\Delta t_{d}$ for the
shorter baselines, it can also be removed owing to its frequency dependence at
the cost of increasing the effective VLBI timing noise. While the timing noise
on the group delay is shown for the case of no differential dispersion, we
find that the delay from the absolute phase when fitting out dispersion is
half this value (see Appendix B for discussion). Similarly the Shapiro time
delay can be removed with a six satellite configuration, as this adds the two
additional constraints needed to remove it using its quadrupolar shape.

Figure 3 also shows how well cosmic distance could be constrained by the
proposed experiment as a function of the typical baseline length. The
baselines over which the solid lines representing $\Delta t_{d}$ have a thick
line style indicate those for which the VLBI timing and scattering delays are
each $<3\%$ of $\Delta t_{d}$. Here we are using $\sigma_{t,\rm group}^{\rm
VLBI}$ and not half this value as we find is possible. For $\nu=4\,$GHz (top
panel), the fiducial scattering delay is important and so limits $3\%$
precision to baselines of $>20$ AU for $d=100~{}$Mpc and $>70$ AU for
$d=1000~{}$Mpc. For some sightlines the scattering will be lower and hence a
higher precision possible. At $8\,$GHz the scattering timescales are much
smaller, and a $3$% measurement of $\Delta t_{d}$ is possible for somewhat
smaller baselines compared to $4\,$GHz.

Both scattering and dispersion become less problematic with increasing
frequency, whereas Shapiro time delays are frequency independent and limit the
precision at longer baselines in a manner for which there is no benefit from
increasing the baseline length unless they are removed. For all baseline
lengths, Shapiro delays limit $3\%$ precision to sources within $d\approx
1000~{}$Mpc. The Shapiro time delay becomes the dominant noise for projected
baselines of $x_{\perp}\gtrsim 100\,$AU.111111Our simple calculation for the
cosmological Shapiro delay does not include redshift evolution in the matter
field, which likely leads to additional inaccuracy for $d\gtrsim 3000\,$Mpc.

Figure 3: Time delays versus projected baseline length $x_{\perp}$ for
$\nu=4\,$GHz (top panel) and $\nu=8\,$GHz (bottom panel). The solid lines are
the cosmological delay owing to wave front curvature, $\Delta t_{d}$, for
$d=100\,$Mpc (blue) and $d=1000\,$Mpc (red). The other lines are estimates for
different systematics: the differential dispersion from the Galaxy (Equation
14 assuming SM$\equiv C_{n}^{2}L=10^{-4}\;\textrm{kpc~{}m}^{-20/3}$), the
scattering time (taken to be $\sigma_{t}^{\rm
sc}=50\,\text{ns}\,[\nu/1\,\text{GHz}]^{-4}$, $2\times$ larger than the mean
NE2001 model at $b=90^{\circ}$; Equation 12), the Shapiro time delay for
$d=10^{3}$Mpc (assuming the cosmological delay is dominant; Equation D4), and
the VLBI group delay timing error (assuming SNR$=5$ and $\Delta\nu=0.2\,\nu$;
Equation 11). Even though it is often larger than the signal $\Delta t_{d}$,
differential dispersion can be removed using its frequency dependence (§ 4.2).
The baselines over which the solid lines representing $\Delta t_{d}$ are
thickest show where VLBI timing and scattering delays (all calculated in the
manner described above) are each $<3\%$ of $\Delta t_{d}$. Shapiro delays, if
not removed using their quadrupolar shape, limit $<3\%$ precision to
$d\lesssim 10^{3}~{}$Mpc.

Figure 4: Estimated fractional distance error, $\sigma_{d}/d$, to a _single_
FRB as a function of distance $d$, for observations at $\nu=4\,$GHz (top
panel) and $\nu=8\,$GHz (bottom panel). The different colors indicate
different detector separations $x_{\perp}$: 12.5 AU in brown, 25 AU in orange,
50 AU in green, 100 AU in red, and 200 AU in purple. The solid curves assume
to compute the timing error ($\sigma_{t}$) that each baseline vector is
constrained to a 1$\sigma$ error of $\delta x=3\,$cm in the top and $\delta
x=1\,$cm in the bottom panel – an error which includes the VLBI timing error,
in addition to including the noise from scattering and Shapiro time delays
assuming the same specifications as used in Figure 3. The highlighted regions
show how these uncertainties change over the specified range of $\delta x$.
The dashed curves, only shown for the three longest baselines, are the same
but assume the Shapiro delay is removed owing to its different angular
dependence. The dotted curve is the fractional redshift uncertainty that
arises if the peculiar velocity can be corrected for using large-scale
structure observations to $200\,$km s-1. The vertical lines show the comoving
light-travel distance that corresponds to redshifts of $z=0.05,\,0.5$ and $5$
in a flat $\Lambda$CDM cosmology with $\Omega_{M}=0.3$ and $h=0.7$.

These estimates for the timing noise relative to $\Delta t_{d}$ suggest that
the proposed experiment would make cosmologically interesting distance
measurements with a detector separation of at least $x_{\perp}\approx$ 10 AU.
Figure 4 quantifies this further. It shows estimates for the fractional
distance error, $\sigma_{d}/d$, to a single FRB as a function of their
distance, for observations at $\nu=4\,$GHz (top panel) and $\nu=8\,$GHz
(bottom panel). The fractional distance error is calculated using our estimate
that $\sigma_{d}/d\approx\sigma_{t}/\Delta t_{d}$ when the $N_{A}-1$ equals
the number of constraints, where $\sigma_{t}$ is determined by adding in
quadrature the scattering, Shapiro, and position+VLBI timing error $\delta
x/c$. While the former two errors assume same specifications as described for
Figure 3, we have combine the VLBI timing error and detector positional errors
into the total geometric error $\delta x$. For $\nu=4\,$GHz we have chosen
$\delta x=3~{}$cm and for $\nu=8\,$GHz we have set $\delta x=1~{}$cm, and we
note that the VLBI absolute phase timing error once marginalizing over
dispersion is given by $c\,\sigma_{t,\rm group}^{\rm
VLBI}/2=0.4\,\textrm{cm}\times(1~{}\textrm{GHz}/\Delta\nu_{\rm
RMS})(5/\textrm{SNR})$ and the effective timing error decreases when more than
four detectors are used owing to the increase in baselines.

The different curves indicate different detector separations $x_{\perp}$: 12.5
AU in brown, 25 AU in orange, 50 AU in green, 100 AU in red, and 200 AU in
purple. The highlighted regions vary $\delta x$ over a range of $1-10~{}$cm
for $\nu=4\,$GHz and $0.5-2~{}$cm for $\nu=8\,$GHz. Without Shapiro delay
removal, four detectors with roughly these separations are required to reach
the quoted precision (solid curves), and six are required for Shapiro delay
removal (dashed curves).121212The VLBI timing error on the more optimistic
six-detector configuration would be improved by a factor of $\approx 2.5$ if
Shapiro delays are not fit owing to the increase in visibilities so that
$\sigma_{d}/d=\sigma_{t}/\Delta t_{d}$ overestimates the error (§ 2). Ignoring
the Shapiro delays we find is appropriate at $x_{\perp}\lesssim 100~{}$AU and
so the distance errors for this case could be even smaller than those quoted
in this section.

The top panel in this figure considers $\nu=4\,$GHz. The orange curves show
our estimate for the error from a constellation with typical baselines of
$x_{\perp}\approx 25\,$AU on a single FRB. This configuration would achieve
$\sigma_{d}/d\approx 0.05$ out to $d=200\,$Mpc for our fiducial choice of
$\delta x=3~{}$cm. A distance of $200\,$Mpc corresponds to approximately
$z=0.05$ in the $\Lambda$CDM cosmology ($\Omega_{M}=0.3$ and $h=0.7$; Appendix
A), the median redshift of the SHOES supernova sample. For the $\nu=8\,$GHz
case shown in the bottom panel, scattering is no longer an important timing
error, allowing $x_{\perp}\approx 25\,$AU to achieve $\sigma_{d}/d\approx 0.1$
out past $d=1000\,$Mpc for $\delta x=1~{}$cm, with a 1% measurement to
$d=200\,$Mpc. We emphasize that these forecasts are for a _single_ FRB.

Again considering our fiducial $\delta x$ and moving to a constellation with
$x_{\perp}\approx 50\,$AU (the green curves), while the explanation for the
trends are the same, at $4\,$GHz now 10% measurements are possible out to
$d=1000\,$Mpc, with better than a $2\%$ precision for $d=200~{}$Mpc. At
$8\,$GHz with again $x_{\perp}\approx 50\,$AU, 10% measurements are possible
out to either 1000 Mpc and $10000$ Mpc, depending on whether Shapiro delays
are not or are removed. At $x_{\perp}\approx 100\,$AU (the red curves), the
errors at small $d$ improve by yet another factor of $(50/100)^{2}$ owing to
the larger $\Delta t_{d}$, but cosmological Shapiro delay is starting to
become the limiting factor at large $d$ for the solid curves. Finally, for
$x_{\perp}\approx 200\,$AU, Shapiro delay cleaning is becoming essential as
there is little improvement over the $x_{\perp}\approx 100\,$AU case without
cleaning. Once the Shapiro delay is removed (as for the dashed curves), then
sub-percent measurements to $10^{4}~{}$Mpc ($z\approx 5$) in the concordance
cosmology are possible at $8\,$GHz. Thus, the scaling $\Delta t_{d}\propto
x_{\perp}^{2}/d$ results in a fast increase in the distance reach with
increased average baseline length.

The dotted curve in Figure 4 is the fractional redshift uncertainty that
arises if the peculiar velocity contribution to the redshift can be corrected
using large-scale structure observations to $\delta v=200\,$km s-1, about half
of the $z=0$ RMS value for the line-of-sight component and in line with the
claimed removal in cosmological analyses of Type 1a supernovae (Peterson et
al., 2022). The fractional redshift uncertainty is calculated as $\delta
v\times(1+z)/[c\,z]$. While for certain configurations the distance can be
constrained to higher precision, there is little gain in doing so below this
peculiar velocity limit.

All of the previous results are on a per-source and per-burst basis.
Individual Type Ia supernova distances only constrain distance to $\approx
10\%$, requiring about $100$ supernova for the current 1% measurement of
$H_{0}$ (Riess et al., 2022). Analogous statistical improvements using
multiple FRBs can of course be done here. Furthermore, repeated observations
of repeating FRBs will offer a consistency check, especially since the
scattering time and differential dispersion to each detector change on month
to year timescales.

##  8 Other potential science

Additionally, we have identified several other potential sciences to which the
proposed experiment could contribute:

the mass distribution in the outer Solar System

The unaccounted gravitational accelerations of the satellites that lead to
$\gtrsim 1\,$cm displacements owes to asteroids with radii of $R\gtrsim
10\,(v/10\,\text{km s}^{-1})^{2/3}~{}$km (§ 6). Such displacements are
potentially detectable if non-gravitational accelerations can be isolated.
Additionally, in a toy picture where mass is spherically distributed, an
excess mass of just $10^{-7}(r/100\,{\rm AU})^{-1/2}M_{\earth}$ interior to an
orbiting satellite results in a potentially measurable centimeter displacement
in one year. Similarly, this experiment may also be sensitive to dark matter
clumps with masses $\gtrsim(200\,\text{km/s})^{2}\times
1\text{cm}/G=10^{-6}\,M_{\earth}$ (similar to the anticipated dark matter mass
within a $100\,$AU cube in the Solar neighborhood), which some dark matter
models predict should be passing near the Solar System with typically a much
larger velocity of $\sim 200\,$km s-1 compared to bound outer Solar System
bodies.

geometric distances to and radio emission from Milky Way pulsars

As our system is capable of measuring distances out to hundreds of
megaparsecs, from a geometric standpoint it can easily measure distances (even
with $0.1~{}$AU baselines) to sufficiently bright pulsars at all distances in
the Milky Way, with the caveat that individual pulses from most pulsars are
difficult to detect with 10 m dishes especially at the higher frequencies at
which the pulsars have $\lesssim\Delta t_{d}$ scattering times as would be
required. Such precise distances would enhance the gravitational wave
sensitivity of pulsar timing arrays (Boyle & Pen, 2012) and even enable pulsar
timing arrays to measure $H_{0}$ (McGrath et al., 2022). Furthermore, the
proposed experiment could shed light on the nature of pulsar emission. It
could resolve their radio emission, as a baseline’s spatial resolution is
$\ell=10^{2}\,\text{km}~{}(x_{\perp}/100\,\text{AU})^{-1}(\nu/5\,\text{GHz})^{-1}(d/1~{}\text{kpc})$.
The nature of pulsar radio emission is still debated, with two classes of
models (e.g. Philippov & Kramer, 2022). The first predicts that the emission
is from the inner magnetosphere at radii of $10-100~{}$km. A $100~{}$AU
baseline has the potential to resolve this component, especially for the
closest pulsars. The other class of emission occurs at the $\sim
5000\,(\tau/0.1\,s)\,$km light cylinder, where $\tau$ is the spin period,
which would be resolved with $x_{\perp}\gtrsim
10\,(\tau/0.1~{}s)^{-1}(\nu/1\,\text{GHz})^{-1}~{}$AU.

the spectrum of density fluctuations from both ISM electrons and the dark
matter

The dispersion delay differences between different satellites should be easily
detectable (§ 4.2). Such measurements would constrain the spectrum of ISM
density fluctuations on the Solar System-scale of our antenna separations,
complementing measurements on smaller and larger scales. This would be
interesting for measuring the spectrum of turbulence (Armstrong et al., 1981)
and also for testing the picture of electron inhomogeneities from $\sim
1~{}$AU current sheets invoked to explain some aspects of scattering and
scintillation (Goldreich & Sridhar, 2006; Pen & Levin, 2014). For some
sightlines the ISM scattering to each antenna would also be constrained,
further testing these models. Our calculations also suggest that a six
satellite system with $x\gtrsim 100~{}$AU would measure the quadrupolar
moments of the Shapiro time delay. In the standard cosmology, these moments
are most sensitive to the abundance and profiles of $10^{10}-10^{13}M_{\odot}$
dark matter halos, but they can be shaped by much smaller structures in
clumpier dark matter models (Appendix D).

micro-Hertz gravitational waves

Timing signals between satellites is sensitive to gravitational wave strains
with frequencies $\lesssim 10^{-4}(x/20\,\text{AU})^{-1}$ Hz. Strains of
$10^{-18}-10^{-16}$ around such frequencies are predicted from various
Galactic and cosmological sources (Sesana et al., 2021). While a strong
science case exists for the ‘midband’ frequencies between the nano-Hertz
frequencies measured by pulsar timing and the $10^{-4}-1\,$Hertz frequencies
probed by LISA (Amaro-Seoane et al., 2017), they unfortunately are difficult
to target with the traditional laser ranging in the inner Solar System
(Fedderke et al., 2022). At our long baselines, radio ranging is likely the
more practical approach because the SNR in the match-filtered electric field
scales inversely with the radiometer separation compared to the quadratic fall
off in SNR because of the dilution of laser photons (and the laser power
requirements are already challenging for 1 AU baselines and reasonable mirror
sizes; Sesana et al. 2021; Fedderke et al. 2021). Here we consider detecting
gravitational waves by measuring baseline distances rather than the less-
ambitious Doppler ranging that has been applied to outer Solar System
spacecraft (Armstrong, 2006). How well the carrier wave phase can be timed
between two of our satellites is limited by the frequency stability of the
onboard atomic clock on at least one satellite (as the signal from other
satellites can be phased to the incoming signal), with a stability
characteristic of the Deep Space Atomic Clock allowing sensitivity to
millimeter displacements for gravitational waves with strain errors of $\delta
h=10^{-15}(\nu_{\rm GW}/10^{-5}~{}\text{Hz})^{-1/2}(x/100\,{\rm
AU})^{-1}W^{-1}$, where $W$ is a window function that is near unity for
gravitational wave frequencies of $\nu_{\rm GW}\sim c/x=2\times
10^{-5}(x/100\,{\rm AU})^{-1}\;$Hz and scales as $W\sim\nu_{\rm GW}x/c$ at
lower $\nu_{\rm GW}$ (since $x$ spans only a fraction of a wavelength). The
best atomic clocks in the world have been improving rapidly and would best the
timing precision of the Deep Space Atomic Clock by five orders of magnitude.
Once no longer limited by the onboard clock, the precision that the carrier
wave can be timed as well as stochastic sources of gravitational acceleration
likely set the error. The precision that the gravitational wave strain can be
measured when limited by timing of the carrier phase to accuracy $\delta
t_{c}$ is (Misra & Enge, 2012)

$\displaystyle\delta h=\frac{c\,\delta\,t_{c}}{x}\,W^{-1}$ $\displaystyle\sim$
$\displaystyle\frac{c}{2\pi\nu x}\sqrt{\frac{\nu_{\rm GW}kT_{\rm sys}}{P_{\rm
rec}}}\,W^{-1},$ (29) $\displaystyle=$ $\displaystyle 2\times
10^{-20}\,W^{-1}\left(\frac{5\,\textrm{GHz}}{\nu}\right)^{2}\left(\frac{10\,\textrm{m}}{D}\right)^{2}\sqrt{\frac{\nu_{\rm
GW}}{10^{-5}\,\text{Hz}}\frac{100\,\text{Watt}}{P_{\text{ant}}}\frac{T_{\text{sys}}}{20\,\text{K}}},$
(30)

where this assumes continuous tracking over the full wave period.
Gravitational accelerations on the orbital timescales of bodies likely sets
the redward limit to $\nu_{\rm GW}\gtrsim 0.01\,\mu$Hz assuming that the non-
gravitational accelerations can be isolated.

##  9 Conclusions

We have described a new method for measuring cosmological distances using the
differences in arrival times to cosmological sources as measured by extremely
long baseline interferometry. In order to vastly improve future measurements
of the late-time expansion of the Universe, we think that the proposed
experiment may be our best option. Despite the huge number of potentially
complicating effects, the proposed experiment seems possible for a
surprisingly well-defined set of mission specifications:

  * •

FRBs are the one source class that is bright enough and sufficiently compact
for this measurement.

  * •

We showed that once baselines reach $x=25$ AU, individual FRBs can be used to
measure percent-level distances to $d=200\,$Mpc ($z=0.05$), and a ten percent
measurement to $d=2000\,$Mpc ($z=0.5$). Going up in scale to $x\approx
100~{}$AU, 1% measurements on the distance are possible for a $z=1$ FRB. In
the limit where the detector space-time positions are known, the primary
limitation on the timing precision for baselines of $x<100\,$AU is Milky Way
ISM scattering at $\lesssim 3\,$GHz and differential dispersion-marginalized
VLBI timing errors at higher frequencies, resulting in a timing noise of
$\sigma_{t}\sim\nu^{-1}$ at $\nu\gtrsim 3\,$GHz.

  * •

The experiment is best at $\nu\gtrsim 3\,$GHz to minimize Milky Way
scattering. A wide bandwidth with $\Delta\nu/\nu\gtrsim 0.1$ (or multiple
bands) is helpful for signal-to-noise considerations and to subtract relative
delays from dispersion based on their frequency dependence, delays we find can
be comparable to the geometric delay of interest. The requirement to use a
higher frequency band than the $\nu\sim 1\,$GHz targeted by most FRB surveys
is perhaps the biggest uncertainty in this proposal, as the FRB population is
less studied there. For baselines of $x\gtrsim 100\,$AU, the curvature in the
total Shapiro time delay limits the precision, and we showed the cosmological
contribution to this delay is typically larger than the Milky Way contribution
from stars. The quadrupolar shape of the contaminating Shapiro delay allows it
to be removed at the cost of two additional satellites.

  * •

The satellite positions need to be determined to several centimeters for our
forecasts to apply, with the error increasing linearly with the positional
uncertainty once above this specification. We showed that GNSS-like direct
trilateration using the dishes themselves appears feasible, although
calibration may also be possible using only distant FRBs themselves. The
latter calibration strategy likely requires a precise on-board accelerometer
to eliminate non-gravitational accelerations and, thus, extend the timescale
needed between calibrations. We estimate that gravitational accelerations from
closely passing asteroids should lead to a $1\,$cm error in satellite
ephemeris over a time of months at the very minimum, but likely much longer.
Without a precise accelerometer, ephemeris calibration must be performed
weekly to account for accelerations from variations in the Solar irradiance,
dust collisions, gaseous drag, and possibly Lorentz forces acting on the
satellites. For weekly calibration, the satellites’ clock accuracy requirement
is in line with the demonstrated accuracy of NASA’s Deep Space Atomic Clock.

  * •

The increasing catalog of FRB sources known to repeat provides a list of
targets for which many are likely to yield an FRB after just hours of
integration. The simplest approach would be for the satellites to point to
such a source and store the voltage time series, telemetering it back to Earth
if prompted that a burst occurred. A more ambitious approach that would
require substantial onboard computation is for each satellite to search the
voltages and identify FRBs on the fly. Then, 10 m diameter dishes are
necessary with tens of beams to detect tens of FRBs a year, although there is
substantial uncertainty in how the observed $\sim 1\,$GHz FRB rates
extrapolate to $\gtrsim 3\,$GHz. Possibilities that could greatly increase the
detection rates are (1) using a more sensitive terrestrial radio telescope for
a template of the electromagnetic waveform, and (2) employing phased elements
in each satellite rather than a single dish to simultaneously observe much of
the sky. A 10 m diameter instrument is also convenient for ephemeris
calibration by direct trilateration, requiring only $\sim 10\,$Watt broadcasts
between antennas separated by $100\,$AU for $\sim 0.1~{}$ns timing using a
modulating $10\,$MHz code and minute integration times. A fold-out design (as
done for the $D=10\,$m Spektr-R satellite that had a 30 $R_{\earth}$ apogee)
could allow a radio dish to be deployed in a modest payload.

  * •

We favor unbound orbits where the baselines gradually drift to larger and
larger separations. As an example, the New Horizons spacecraft achieved a
terminal velocity of $10\,$km s-1 via gravity assists off of planets, enabling
it to travel $40~{}$AU in a decade. Unbound orbits would allow nailing down
science that can be accomplished at shorter baselines first and a quadratic-
with-time increase in the size of $\Delta t_{d}$, and hence the precision of
distance determinations, as the detectors drift apart. It might even be
possible to launch all the antennas in one rocket and disperse them in
different directions by scattering off a Solar System body.

Some studies have discussed the possibility of detecting the parallaxes of
cosmological objects (e.g. Ding & Croft 2009; Chakrabarti et al. 2022). The
proposed experiment to detect wave front curvature can be thought of in terms
of parallax with instruments separated by $\sim x_{\perp}$ and where the
angular resolution is set by $\sim\lambda/x_{\perp}$ (§ 2). If we take our
fiducial specifications of $x_{\perp}=100~{}$AU and $\nu=5$ GHz, such an
experiment’s angular resolution is more than five orders of magnitude more
precise than the state-of-the-art $\sim 100~{}\mu$-arcsecond astrometric
localizations that the Gaia satellite achieves toward the brightest quasars
(Gaia Collaboration et al., 2016). Hence the effective sensitivity of our
method for measuring distance is improved by a similar factor (times the
additional factor of $\sim x_{\perp}/(2~{}\text{AU})$ when comparing to annual
parallax).

We also identified several other interesting sciences that could be
accomplished with the proposed instrument (§8). These include measuring the
distance and resolving the radio emission region of Galactic pulsars,
constraining the mass distribution in the outer Solar System at the millionths
of Earth mass-level (interesting both for Solar System and dark matter
science), direct constraints on the density distribution in the interstellar
medium (ISM) on $x_{\perp}$ scales, and potentially interesting sensitivities
to $\sim 0.01-100\,\mu$Hz gravitational waves. The proposed instrument not
only would provide a more direct means to constrain distance in cosmology, but
also contribute to answering other key questions.

We thank Yakov Faerman, Jason Hessels, Bryna Hazelton, Leonid Gurvits, Kiyoshi
Masui, Casey McGrath, Miguel Morales, Ue-Li Pen, Alexander Philippov,
Alexander Tchekhovskoy, and Huangyu Xiao for useful discussions and comments.
We thank the two anonymous referees for valuable feedback. MM acknowledges
support from NSF award AST-2007012. KB acknowledges support from the DiRAC
Institute in the Department of Astronomy at the University of Washington. The
DiRAC Institute is supported through generous gifts from the Charles and Lisa
Simonyi Fund for Arts and Sciences and from the Washington Research
Foundation.

##  Appendix A FRW spacetime derivation

In flat space, the distances that appear in the main text are really the
effective distance from the accumulated phase in the wave fronts. Namely,

$d=\lambda_{\rm obs}\int_{t_{\rm em}}^{t_{\rm
obs}}\frac{c\,dt}{\lambda}=\int_{0}^{z}\frac{c\,dz}{H(z)},$ (A1)

and similarly for $d_{x}$ (c.f. Figure 2), where $t_{\rm em}$ and $t_{\rm
obs}$ are, respectively, the emission and observation times and
$\lambda=\lambda_{\rm obs}/(1+z)$ relates the wavelength to its observed
(redshifted) value. These reduce to the Euclidean distance at low redshift.
The rightmost integral in Equation A1 is just the comoving light-travel
distance. Thus, one is free to draw the antenna–antenna–FRB triangle in Figure
2 at the present time, ignoring the time dependence of FRW expansion. The same
result is well established for lensing time delays (Narayan & Bartelmann,
1996).

To generalize our calculation to curved FRW space-times, Figure 2 should be
redrawn as a triangle in an open or closed geometry and, rather than using the
law of cosines to calculate $d_{x}$ (c.f. Equation 1), we use these spaces’
generalization:

$\displaystyle\cos(\theta_{d_{x}})$
$\displaystyle=\cos(\theta_{d})\cos(\theta_{x})+\sin(\theta_{d})\sin(\theta_{x})\cos(\angle_{d-x}),$
(A2)
$\displaystyle\approx\cos(\theta_{d})(1-\theta_{x}^{2}/2)+\sin(\theta_{d})\theta_{x}\cos(\angle_{d-x}),$
(A3)

where $\angle_{d-x}$ denotes the angle adjoining the line segments $d$ and $x$
in the triangle and $\theta_{r}\equiv kr/R$, with $r$ as a stand-in for other
distance variables. The value $k=1$ corresponds to a closed geometry and $k=i$
to an open one, where $R^{-1}\equiv\sqrt{|\Omega_{k}|}H_{0}/c$ is the present-
day radius of curvature.131313These relations for the law of cosines can be
found on Wikipedia, with references dating into the nineteenth century. The
identification with the curvature space density $\Omega_{k}$ for FRW
cosmologies is given in many cosmology textbooks (e.g. Peacock, 1999).
Equation A3 takes the applicable limit that $\theta_{x}$ is much less than
one. To further simplify, we note that
$\cos(\theta_{d_{x}})\approx\cos(\theta_{d})-\sin(\theta_{d})k(d_{x}-d)/R-\cos(\theta_{d})[k(d_{x}-d)/R]^{2}/2$.
An equation quadratic in $d_{x}-d$ results, which solving yields the root:

$\displaystyle c\,\Delta t$ $\displaystyle=$ $\displaystyle
d_{x}-d=\frac{R}{k}\left[-\tan(\theta_{d})+\sqrt{\tan(\theta_{d})^{2}+\frac{k^{2}x^{2}}{R^{2}}-2\tan(\theta_{d})\frac{kx}{R}\cos(\angle_{d-x})}\right],$
(A4) $\displaystyle\approx$
$\displaystyle-x\cos(\angle_{d-x})+\frac{kx^{2}\sin(\angle_{d-x})^{2}}{2R\tan(\theta_{d})}+{\cal
O}\left(\frac{x^{3}}{R^{2}\tan(\theta_{d})^{2}}\right),$ (A5) $\displaystyle=$
$\displaystyle-{\boldsymbol{x}}\cdot{\boldsymbol{\widehat{d}}}+\frac{kx_{\perp}^{2}}{2R\tan(\theta_{d})}+{\cal
O}\left(\frac{x^{3}}{R^{2}\tan(\theta_{d})^{2}}\right),$ (A6)

where the second line reapplies the limit $x/R\ll 1$ and $x_{\perp}\equiv
x\sin(\angle_{d-x})$. Equation A6 shows that a measurement of the time delay
between local detectors is sensitive to the cosmological distance
$R\tan(\theta_{d})$. This distance is similar to the luminosity and angular
diameter distances that many cosmological observables are sensitive to, which
are given by $R\sin(\theta_{d})$ up to factors of $(1+z)$.

In conclusion, the curved-space generalization of the time delay owing to
curvature of the wave front is

$\Delta t_{d}\approx\frac{kx_{\perp}^{2}}{2cR\tan(\theta_{d})}.$ (A7)

In the limit that the space’s radius of curvature $R$ is much larger than $d$,
this reduces to our Euclidean result in § 2 but with Equation A1 for the
distance.

We learned after submitting this manuscript to the preprint server that a
nearly identical derivation as above was done in McGrath et al. (2022),
considering the effect of curvature of the wave front of gravitational waves
on pulsar timing arrays. Furthermore, Weinberg (1972) define the distance
$\tan(\theta_{d})$ as the parallax distance. Since wave front timing can be
thought of in terms of parallax (§ 2), that wave front curvature probes the
parallax distance is not surprising.

##  Appendix B fitting for the dispersion

Here we calculate how well the effects of dispersion to each detector can be
fit for and removed. We again use the notation that $\Delta t$ is our
parameter for the geometric delay, and we adopt $\Delta\tau_{d}$ their
differential dispersive delay. We want to find the delays
$\boldsymbol{p}=(\Delta t,~{}\Delta\tau_{d})$ that minimize
$\chi^{2}=\sum_{i}|{\cal V}_{i}-{\cal V}_{i,\rm obs}|^{2}/\sigma_{i}^{2}$,
where ${\cal V}_{i}\equiv\langle E_{1i}E_{2i}^{*}\rangle$ is the complex
correlation of the electric field in frequency channel $\nu_{i}$ between two
detectors (the ‘visibility’) computed by sampling different times across the
FRB, the subscript ‘obs’ indicates the observed, whereas without it is the
modeled, and $\sigma_{i}$ is the channel noise on ${\cal V}_{i,\rm obs}$.
Critically, since a time delay results in a phase in Fourier space, delays
alter ${\cal V}_{i}$ by giving it a complex phase of $\exp[i2\pi\nu_{i}(\Delta
t+\Delta\tau_{d}(\nu_{i}/\bar{\nu})^{-2})]$, where $\bar{\nu}$ is the SNR-
weighted midpoint of the band. The Fisher information matrix (which describes
the curvature of the log likelihood) is given by
$F_{nm}=1/2\,d\chi^{2}/dp_{n}dp_{m}\approx\sum_{i}\sigma_{i}^{-2}d{\cal
V}_{i}/dp_{n}d{\cal V}_{i}^{*}/dp_{m}$ (e.g. Dodelson, 2003), and the
$1\,\sigma$ uncertainty on each delay parameter is given for $\Delta t$ by
$\boldsymbol{F}^{-1/2}|_{\Delta t\Delta t}$. (We do not need to include the
amplitudes of the visibility as another parameter because they are
uncorrelated with the other parameters – $\boldsymbol{F}$ is diagonal in this
parameter – and so can be fit independently.)

First let us consider the case where we do not know the absolute phase
$\theta_{i}$, defined as the phase at the frequency midpoint of the band
$\bar{\nu}$. The absolute phase can be complicated by clock phase errors – and
for terrestrial measurements by the atmosphere – and also the space-time
position of the detectors must be known to better than a wavelength for
solutions for $\theta_{i}$ to not have a phase-wrapping degeneracy. In this
case, we rewrite the complex visibility phase as
$\exp[i\\{\theta_{i}+2\pi(\nu_{i}-\bar{\nu})(\Delta
t+\Delta\tau_{d}(\nu_{i}/\bar{\nu})^{-2})\\}]$. As $\Delta t$ only includes
the phase across the band and the rest is absorbed into $\theta_{i}$, this
case is measuring the equivalent of the group delay over the band
($d\phi/d\nu$ when $\Delta\tau_{d}=0$). This yields the VLBI group delay
timing noise $\sigma_{t,\rm group}^{\rm VLBI}$ given by Equation 11 in the
case where differential dispersion is not a parameter so that $F$ is a scalar.
(We do not need to consider $\theta_{i}$ as a parameter as referencing $\Delta
t$ to the effective center of the band makes this parameter independent; see
Appendix 12.1 in Thompson et al. (2017).) One can check that it does, noting
that $d{\cal V}_{i}/d\Delta t\approx 2\pi i(\nu-\bar{\nu})\,{\cal V}_{i}$. If
we then sum over all frequency channels assuming the signal to noise is
channel independent, this yields $F=[2\pi\,\Delta\nu_{\rm RMS}{\rm SNR}]^{2}$,
where ${\rm SNR}^{2}=\sum_{i}|{\cal V}_{i}|^{2}/\sigma_{i}^{2}$ is the total
signal-to-noise ratio on the intensity that the baseline measures. This agrees
with Equation 11, except there we wrote there the more general expression for
when the channel signal-to-noise ratio varies with frequency. In other
references this result is derived by least squares fitting directly to the
phase (e.g. Rogers, 1970; Thompson et al., 2017).

Now including dispersion in this group velocity limit where we are only using
the phase trends across the band, this becomes a three parameter Fisher matrix
calculation as $\theta_{i}$ is weakly correlated with $\Delta\tau_{d}$. In
this case, the error becomes significantly larger, $\approx
10\,\sigma_{t,\,\rm group}^{\rm VLBI}(0.2\,\nu/\Delta\nu)$. For
$\Delta\nu/\nu\gtrsim 0.1$, interesting cosmological constraints may still be
possible with such timing noises (§ 7).

However, there is no reason to throw away the absolute phase information as
the design of the proposed experiment requires precise baseline measurements.
When using the full phase, the Fisher matrix becomes
$\boldsymbol{F}\propto[\int_{\Delta\nu}\nu^{2}\,d\nu~{}~{}~{}\bar{\nu}^{2}\Delta\nu;~{}\bar{\nu}^{2}\Delta\nu~{}~{}\bar{\nu}^{4}\int_{\Delta\nu}\nu^{-2}\,d\nu]$.
One can show that the error on the dispersion-marginalized phase velocity is
exactly $\sqrt{[\boldsymbol{F}^{-1}]_{0,0}}=\sigma_{t,\rm group}^{\rm
VLBI}/2=[4\pi\Delta\nu_{\rm RMS}{\rm SNR}]^{-1}$. The delay from the absolute
phase is subject to a $2\pi$ phase-wrapping degeneracy. The correct phase can
be distinguished with multiple frequencies or a broad band. The $n^{\rm th}$
phase fringe can be distinguished from the zeroth at $1\sigma$ once
$\nu/\Delta\nu\lesssim n\,{\rm SNR}$.

Global navigation systems often use a second frequency band to remove
ionospheric delays. Their signal template is perfectly known, in contrast to
the noisy ‘template’ supplied by the second VLBI receiver. However, the
results carry over: Our dispersion-marginalized noise estimate in the narrow
two-channel limit increases the timing noise by the factor
$(\nu_{1}^{2}+\nu_{2}^{2})/(\nu_{1}^{2}-\nu_{2}^{2})$ over the timing noise in
either channel in the absence of dispersion, reproducing the result known for
two-channel GNSS (Misra & Enge, 2012).

We find that simultaneously fitting for a term $\propto\nu^{-4}$ to remove
weak refractive scattering (§ 4.1) comes at a larger cost. When using the
absolute phase, the error is increased to $\approx 5\,\sigma_{t,\rm
group}^{\rm VLBI}(\Delta\nu/0.2\,\nu)$. However, since we favor targeting
frequencies where the scattering time delays are $\lesssim\nu^{-1}$, such
marginalization is unnecessary to reach the timing goal of
$\sigma_{t}\sim\nu^{-1}$.

The above assumes that the dispersion is fit for each visibility
independently. One can also fit for the geometric delay $\Delta t_{d}$ and the
$N_{A}-1$ differential dispersion delays simultaneously. This approach incurs
a similar cost for marginalizing out dispersion, since dispersion is
distinguished from geometric time delays via its frequency dependence.

##  Appendix C Dispersion delays from ISM turbulence

Here we estimate the delays from ISM turbulence leading to varying electron
columns along the sightlines to the different detectors. The variance of the
differences in dispersive delays $\tau_{d}$ between pairs of sightlines is
given by

$\displaystyle\textrm{Var}\left[\tau_{d,1}-\tau_{d,2}\right]$
$\displaystyle=2\left\langle\Delta\tau_{d,1}^{2}\right\rangle-2\left\langle\Delta\tau_{d,1}\Delta\tau_{d,2}\right\rangle,$
(C1)

where we write the difference in the dispersion time delay of the sightline
with path $P_{i}$ relative to the mean value as $\Delta\tau_{d,i}$, angular
brackets indicate an ensemble average, and

$\displaystyle\left\langle\Delta\tau_{d,1}\Delta\tau_{d,2}\right\rangle$
$\displaystyle=\kappa^{2}\left\langle\int_{P_{1}}dx_{1}\Delta
n_{e}({\boldsymbol{x}}_{1})\int_{P_{2}}dx_{2}\Delta
n_{e}({\boldsymbol{x}}_{2})\right\rangle,$ (C2)

and $\Delta n_{e}$ is the 3D field of electron density fluctuations. Replacing
$\Delta n_{e}$ with its Fourier transform, the expectation value becomes

$\displaystyle\left\langle\Delta\tau_{d,1}\Delta\tau_{d,2}\right\rangle=$
$\displaystyle\kappa^{2}\int_{P_{1}}dx_{1}\int_{P_{2}}dx_{2}\iint\frac{d^{3}k_{1}d^{3}k_{2}}{(2\pi)^{6}}\left\langle\Delta
n_{e}({\boldsymbol{k}}_{1})\Delta n_{e}({\boldsymbol{k}}_{2})^{*}\right\rangle
e^{-i({\boldsymbol{k}}_{1}\cdot{\boldsymbol{x}}_{1}-{\boldsymbol{k}}_{2}\cdot{\boldsymbol{x}}_{2})},$
(C3) $\displaystyle=$
$\displaystyle\kappa^{2}\int_{P_{1}}dx_{1}\int_{P_{2}}dx_{2}\int\frac{d^{3}k}{(2\pi)^{3}}P_{e}(k)e^{-i{\boldsymbol{k}}\cdot({\boldsymbol{x}}_{1}-{\boldsymbol{x}}_{2})},$

where we have used the property of real fields that $\Delta
n_{e}({\boldsymbol{k}})^{*}=\Delta n_{e}(-{\boldsymbol{k}})$ and defined the
electron density power spectrum as $\left\langle\Delta
n_{e}({\boldsymbol{k}}_{1})\Delta
n_{e}({\boldsymbol{k}}_{2})^{*}\right\rangle=(2\pi)^{3}P_{e}(k_{1})\delta^{D}({\boldsymbol{k}}_{1}-{\boldsymbol{k}}_{2})$.
Evaluating the integral over the line-of-sight wavenumber along $P_{1}$ gives
a $\delta$-function that eliminates the light of sight wavenumber integral. We
next evaluate the spatial line of sight integral along $P_{2}$ to a length of
$L$, representing the size of the region containing electron density
fluctuations (i.e. the extent of the Milky Way ISM), yielding

$\displaystyle\left\langle\Delta\tau_{d,1}\Delta\tau_{d,2}\right\rangle=\kappa^{2}L\int\frac{d^{2}k_{\perp}}{(2\pi)^{2}}P_{e}(k)e^{-ik_{\perp}\cdot({\boldsymbol{x}}_{\perp,1}-{\boldsymbol{x}}_{\perp,2})}=\kappa^{2}L\int_{0}^{\infty}\frac{dk}{2\pi}kP_{e}(k)J_{0}(kx_{\perp}),$
(C4)

where the last equality assumed that the paths are parallel and separated by a
transverse distance of $x_{\perp}$. We can use this expression to evaluate the
square of the standard deviation in dispersion time delays between the two
lines of sight defined in Equation C1:

$\displaystyle\sigma^{2}\left[\tau_{d,1}-\tau_{d,2}\right]$
$\displaystyle=2\kappa^{2}L\int_{0}^{\infty}\frac{dk}{2\pi}k\,P_{e}(k)\,\left[1-J_{0}(kx_{\perp})\right].$
(C5)

For the Milky Way, the electron density distribution is given by a Kolmogorov-
like power law $P_{e}(k)=(2\pi)^{3}C_{n}^{2}\,k^{-11/3}$ between some $\sim
10^{9}\,$cm inner and $\sim 1\,$pc outer length scales, where the $(2\pi)^{3}$
owes to our Fourier convention as we have adopted the standard definition for
$C_{n}^{2}$ (e.g. Draine, 2011). In the solar neighborhood, measurements find
$C_{n}^{2}\sim 5\times 10^{-17}\textrm{cm}^{-20/3}$ (Armstrong et al., 1981;
Draine, 2011). This integral is not sensitive to the inner or outer length
scales, and evaluating the square root yields the standard deviation

$\displaystyle\sigma\left[\tau_{d,1}-\tau_{d,2}\right]$
$\displaystyle=9.4~{}\kappa\,C_{n}x_{\perp}^{5/6}\sqrt{L},$ (C6)
$\displaystyle=280~{}\textrm{ns}\left(\frac{L}{0.1~{}\textrm{kpc}}\right)^{1/2}\left(\frac{x_{\perp}}{100~{}\textrm{AU}}\right)^{5/6}\left(\frac{\nu}{5~{}\textrm{GHz}}\right)^{-2}\left(\frac{C_{n}^{2}}{5\times
10^{-17}\textrm{cm}^{-20/3}}\right)^{1/2}.$ (C7)

Much of this contribution to $\sigma\left[\tau_{d,1}-\tau_{d,2}\right]$ is
from long-wavelength modes that manifest as a gradient that is degenerate with
the $-{\boldsymbol{x}}\cdot{\boldsymbol{\widehat{d}}}$ delay from the source
location on the sky. Crudely, only the contribution of modes in
$k>\pi/x_{\perp}$ could bias the delay from wave front curvature. These
wavenumbers (and indeed the same holds when restricting to the even more
curvature-like modes with $\pi/x_{\perp}<k<2\pi/x_{\perp}$) contribute only
10% of the integral in Equation C4. Thus, the fraction that is non-planar and,
hence, contaminates our distance measurement is

$\displaystyle\sigma\left[\tau_{d,1}-\tau_{d,2}\right]_{\rm non-planar}\approx
90~{}\textrm{ns}\left(\frac{L}{0.1~{}\textrm{kpc}}\right)^{1/2}\left(\frac{x_{\perp}}{100~{}\textrm{AU}}\right)^{5/6}\left(\frac{\nu}{5~{}\textrm{GHz}}\right)^{-2}\left(\frac{C_{n}^{2}}{5\times
10^{-17}\textrm{cm}^{-20/3}}\right)^{1/2}.$ (C8)

We use this expression for $\sigma_{t}^{\rm disp}$ in § 4.2 (e.g. eqn. 14).

##  Appendix D Cosmological contribution to the Shapiro delay

This appendix calculates in detail the cosmological Shapiro time delay from
large-scale structure. As the Shapiro delay along a sightline can be written
as $\frac{2}{c^{3}}\int_{0}^{d}d\chi\Phi$, where $\Phi$ is the Newtonian
gravitational potential, the Shapiro delay between two sightlines separated by
${\boldsymbol{x}}_{\perp}$ can be calculated via Taylor expansion assuming the
potential is smooth on the scale ${\boldsymbol{x}}_{\perp}$: $\Delta t_{\rm
grav}=\frac{2}{c^{3}}\int_{0}^{d}d\chi({{\boldsymbol{x}}_{\perp}}_{i}{\boldsymbol{\nabla}}_{j}\Phi+\frac{1}{2}{{\boldsymbol{x}}_{\perp}}_{i}{{\boldsymbol{x}}_{\perp}}_{j}\nabla_{i}\nabla_{j}\Phi+...)$.
The quadratic-in-$x_{\perp}$ term inside the integral can be rewritten as
$\frac{1}{2}{{\boldsymbol{x}}_{\perp}}_{i}{{\boldsymbol{x}}_{\perp}}_{j}\frac{\nabla_{i}\nabla_{j}}{\nabla^{2}}4\pi
G\rho$ using Poisson’s equation for the Newtonian gravitational potential and
noting that only long-wavelength modes in the line-of-sight direction
contribute and so we can ignore the line-of-sight derivatives. We can write
the variance of the cosmological component of $\Delta
t_{\text{grav}}^{\text{quad}}$ in terms of the matter overdensity power
spectrum using an approach nearly identical to in § C except the different
angular weighting that appears:

$\displaystyle\text{Var}[\Delta t_{\text{grav}}^{\text{quad}}]$
$\displaystyle=$ $\displaystyle\frac{(4\pi
G\bar{\rho})^{2}d}{c^{6}}\int\frac{d^{2}{\boldsymbol{k}}_{\perp}}{(2\pi)^{2}}\,\frac{({\boldsymbol{k}}_{\perp}\cdot{\boldsymbol{x}}_{\perp})^{4}}{k_{\perp}^{4}}P_{\delta}(k_{\perp})\exp[-i{\boldsymbol{k}}_{\perp}\cdot{\boldsymbol{x}}_{\perp}],$
(D1) $\displaystyle=$ $\displaystyle\frac{(4\pi
G\bar{\rho})^{2}d\,x_{\perp}^{4}}{c^{6}}\int\frac{dk_{\perp}}{(2\pi)}J_{2}(k_{\perp}x_{\perp})k_{\perp}\left(\frac{3}{(k_{\perp}x_{\perp})^{2}}-1\right)\,P_{\delta}(k_{\perp}),$
(D2) $\displaystyle\approx$ $\displaystyle\frac{3(4\pi
G\bar{\rho})^{2}d\,x_{\perp}^{4}}{8c^{6}}\int\frac{dk_{\perp}}{(2\pi)}k_{\perp}P_{\delta}(k_{\perp}),$
(D3)

where we have further assumed no evolution in the statistics of the density
field over the sightline (which should make this inaccurate for $d\gtrsim
3000\;$Mpc, corresponding to $z\gtrsim 1$) and the last approximation uses
that the contribution to the integral owes to structures with
$x_{\perp}k_{\perp}\ll 1$, at least in the standard cosmology. To evaluate the
integral in Equation D3, we adopt the $z=0.1$ HaloFit model for the nonlinear
matter power spectrum (Smith et al., 2003), finding that the standard
deviation of the cosmological Shapiro time delay is

$\sigma_{\text{t, \text{grav}}}^{\rm cosmo}\approx 0.08~{}{\rm
ns}~{}\left(\frac{x_{\perp}}{100~{}\textrm{AU}}\right)^{2}\left(\frac{d}{100~{}\text{Mpc}}\right)^{1/2}.$
(D4)

This is somewhat larger than our estimate for the delay from stars in the
Galactic disk. A significant fraction of this variance can be reduced by
selecting sightlines that do not pass near large dark matter halos. (Half of
the contribution to this integral comes from halo virial radius scales of
$k\in 2\pi/[0.1-1~{}\text{Mpc}]$, and for sightlines with a significantly
enhanced cosmological Shapiro delay the contribution from large halos would be
even larger.) However, measuring this delay would itself be interesting as it
is a probe of nonlinear scales, with an identical wavenumber weighting of
modes as the RMS magnification from gravitational lensing, and so constraining
$\sigma_{\text{t, \text{grav}}}^{\rm cosmo}$ would be very analogous to the
supernova magnification constraints that have placed limits on the clumpiness
of the dark matter (Dodelson & Vallinotto, 2006; Zumalacárregui & Seljak,
2018).

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