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.

European Pulsar Timing Array limits on an isotropic stochastic gravitational-wave background