Are we there yet? Time to detection of nanohertz gravitational waves
  based on pulsar-timing array limits
  Decade-long timing observations of arrays of millisecond pulsars have placed
highly constraining upper limits on the amplitude of the nanohertz
gravitational-wave stochastic signal from the mergers of supermassive
black-hole binaries ($\sim 10^{-15}$ strain at $f = 1/\mathrm{yr}$). These
limits suggest that binary merger rates have been overestimated, or that
environmental influences from nuclear gas or stars accelerate orbital decay,
reducing the gravitational-wave signal at the lowest, most sensitive
frequencies. This prompts the question whether nanohertz gravitational waves
are likely to be detected in the near future. In this letter, we answer this
question quantitatively using simple statistical estimates, deriving the range
of true signal amplitudes that are compatible with current upper limits, and
computing expected detection probabilities as a function of observation time.
We conclude that small arrays consisting of the pulsars with the least timing
noise, which yield the tightest upper limits, have discouraging prospects of
making a detection in the next two decades. By contrast, we find large arrays
are crucial to detection because the quadrupolar spatial correlations induced
by gravitational waves can be well sampled by many pulsar pairs. Indeed, timing
programs which monitor a large and expanding set of pulsars have an $\sim 80\%$
probability of detecting gravitational waves within the next ten years, under
assumptions on merger rates and environmental influences ranging from
optimistic to conservative. Even in the extreme case where $90\%$ of binaries
stall before merger and environmental coupling effects diminish low-frequency
gravitational-wave power, detection is delayed by at most a few years.
