  Quantum fluctuations of the gravitational field in the early Universe,
amplified by inflation, produce a primordial gravitational-wave background
across a broad frequency band. We derive constraints on the spectrum of this
gravitational radiation, and hence on theories of the early Universe, by
combining experiments that cover 29 orders of magnitude in frequency. These
include Planck observations of cosmic microwave background temperature and
polarization power spectra and lensing, together with baryon acoustic
oscillations and big bang nucleosynthesis measurements, as well as new pulsar
timing array and ground-based interferometer limits. While individual
experiments constrain the gravitational-wave energy density in specific
frequency bands, the combination of experiments allows us to constrain
cosmological parameters, including the inflationary spectral index, $n_t$, and
the tensor-to-scalar ratio, $r$. Results from individual experiments include
the most stringent nanohertz limit of the primordial background to date from
the Parkes Pulsar Timing Array, $\Omega_{\rm gw}(f)<2.3\times10^{-10}$.
Observations of the cosmic microwave background alone limit the
gravitational-wave spectral index at 95\% confidence to $n_t\lesssim5$ for a
tensor-to-scalar ratio of $r = 0.11$. However, the combination of all the above
experiments limits $n_t<0.36$. Future Advanced LIGO observations are expected
to further constrain $n_t<0.34$ by 2020. When cosmic microwave background
experiments detect a non-zero $r$, our results will imply even more stringent
constraints on $n_t$ and hence theories of the early Universe.
