European Pulsar Timing Array Limits On An Isotropic Stochastic
  Gravitational-Wave Background
  We present new limits on an isotropic stochastic gravitational-wave
background (GWB) using a six pulsar dataset spanning 18 yr of observations from
the 2015 European Pulsar Timing Array data release. Performing a Bayesian
analysis, we fit simultaneously for the intrinsic noise parameters for each
pulsar, along with common correlated signals including clock, and Solar System
ephemeris errors, obtaining a robust 95$\%$ upper limit on the dimensionless
strain amplitude $A$ of the background of $A<3.0\times 10^{-15}$ at a reference
frequency of $1\mathrm{yr^{-1}}$ and a spectral index of $13/3$, corresponding
to a background from inspiralling super-massive black hole binaries,
constraining the GW energy density to $\Omega_\mathrm{gw}(f)h^2 <
1.1\times10^{-9}$ at 2.8 nHz. We also present limits on the correlated power
spectrum at a series of discrete frequencies, and show that our sensitivity to
a fiducial isotropic GWB is highest at a frequency of $\sim 5\times10^{-9}$~Hz.
Finally we discuss the implications of our analysis for the astrophysics of
supermassive black hole binaries, and present 95$\%$ upper limits on the string
tension, $G\mu/c^2$, characterising a background produced by a cosmic string
network for a set of possible scenarios, and for a stochastic relic GWB. For a
Nambu-Goto field theory cosmic string network, we set a limit
$G\mu/c^2<1.3\times10^{-7}$, identical to that set by the {\it Planck}
Collaboration, when combining {\it Planck} and high-$\ell$ Cosmic Microwave
Background data from other experiments. For a stochastic relic background we
set a limit of $\Omega^\mathrm{relic}_\mathrm{gw}(f)h^2<1.2 \times10^{-9}$, a
factor of 9 improvement over the most stringent limits previously set by a
pulsar timing array.
