Gravitational-Wave Cosmology across 29 Decades in Frequency

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, ${\mathrm{\Omega }}_{\mathrm{GW}}(f)<2.3\times {10}^{-10}$. Observations of the cosmic microwave background alone limit the gravitational-wave spectral index at 95% confidence to $n_t\lesssim 5$ 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 nonzero $r$, our results will imply even more stringent constraints on $n_t$ and, hence, theories of the early Universe.

Popular Summary

The first detection of gravitational waves by the LIGO Scientific and Virgo Collaborations has heralded a new era of gravitational-wave astronomy and cosmology. As with light, there is a broad spectrum of gravitational waves with a variety of observatories and methods used for detecting different wavelengths of gravitational waves. Each observatory is tuned to a particular wavelength. The LIGO/Virgo team measured black holes with masses about 30 times that of our Sun, and pulsar timing arrays search for black holes weighing a billion times more. One gravitational wave source is common to all experiments: quantum fluctuations in the early Universe when it was a fraction of a second old. These quantum fluctuations may have been amplified by a rapid expansion of the Universe known as “inflation” to form a primordial gravitational-wave background. Such a background is a key prediction of inflationary cosmology. Here, we combine data from experiments across 29 orders of magnitude in gravitational-wave frequency, ranging from cosmic microwave background experiments to ground-based interferometers, to constrain the size of these quantum fluctuations and the slope of the primordial gravitational-wave spectrum.

We combine constraints on the gravitational background from the cosmic microwave background, pulsar timing arrays, big bang nucleosynthesis, baryon acoustic oscillations, and ground-based interferometer gravitational-wave experiments. We constrain theoretical models of the expansion history of the early Universe. We establish a new way of doing gravitational-wave cosmology from the combined analysis of many individual gravitational-wave experiments, and we make significant advances in the data analysis of some individual experiments, which we expect will become standard tools. We eagerly await data from the Square Kilometre Array, the Five Hundred Meter Aperture Spherical Telescope, and Advanced LIGO.

Direct detection of gravitational waves from inflation with any of the experiments noted above will engender great excitement and will truly bring our method to the forefront of research.

Figure 1

Experimental constraints on ${\mathrm{\Omega }}_{\mathrm{GW}}(f)$. The black star is the current Parkes Pulsar Timing Array (PPTA) upper limit and all black curves and data points are current 95% confidence upper limits. The gray curve and triangle are, respectively, the predicted aLIGO sensitivity and PPTA sensitivity with 5 more years of data. The indirect GW limits are from CMB temperature and polarization power spectra, lensing, BAOs, and BBN. Models predicting a power-law spectrum that intersect with an observational constraint are ruled out at $>95\%$ confidence. We show five predictions for the GW background, each with $r=0.11$, and with $n_t=0.68$ (orange curve), $n_t=0.54$ (blue curve), $n_t=0.36$ (red curve), $n_t=0.34$ (magenta curve), and the consistency relation, $n_t=-r/8$ (green curve), corresponding to minimal inflation.

Figure 2

Combined, two-dimensional posterior distribution for the tensor-to-scalar ratio $r$, and the blue tilt of the GW spectrum $n_t$, using CMB, PPTA, indirect, and LIGO observations. The contours are the 95% and 99% limits. The green, dashed curve shows the consistency relation, $n_t=-r/8$, while the red and blue triangles correspond, respectively, to the red and blue curves in Fig 1. For clarity, the right-hand panel is a zoomed-in version of the left-hand panel, with additional posterior distributions shown. See Sec. 3a for a description of each posterior distribution.