Using Gravitational Wave Parallax to Measure the Hubble Parameter with
  Pulsar Timing Arrays
  We demonstrate how pulsar timing arrays (PTAs) can, in principle, yield a
purely gravitational wave (GW) measurement of the luminosity distance and
comoving distance to a supermassive black hole binary source, hence providing
an estimate of the source redshift and the Hubble constant. The luminosity
distance is derived through standard measurement of the chirp mass, which for
the slowly evolving binary sources in the PTA band can be found by comparing
the frequency of GW-timing residuals at the Earth compared to those at distant
pulsars in the array. The comoving distance can be measured from GW-timing
parallax caused by the curvature of the GW wavefronts. This can be detected for
single sources at the high-frequency end of the PTA band out to Gpc distances
with a future PTA containing well-timed pulsars out to $\mathcal{O}(10)$ kpc,
when the pulsar distance is constrained to less than a GW wavelength. Such a
future PTA, with $\gtrsim 30$ pulsars with precise distance measurements
between 1 and 20~kpc, could measure the Hubble constant at the tens of percent
level for a single source at $0.1 \lesssim z \lesssim 1.5$. At $z \lesssim
0.1$, the luminosity and comoving distances are too similar to disentangle,
unless the fractional error in the luminosity distance measurement is decreased
below 10%. At $z\gtrsim 1.5$, this measurement will likely be restricted by a
signal-to-noise ratio threshold. Generally, clarification of the different
types of cosmological distances that can be probed by PTAs, and their relation
to pulsar distance measurements is important for ongoing PTA experiments aimed
at detecting and characterizing GWs.
