Constraining the gravitational wave energy density of the Universe using Earth’s ring

The search for gravitational waves is one of today’s major scientific endeavors. A gravitational wave can interact with matter by exciting vibrations of elastic bodies. Earth itself is a large elastic body whose so-called normal-mode oscillations ring up when a gravitational wave passes. Therefore, precise measurement of vibration amplitudes can be used to search for the elusive gravitational-wave signals. Earth’s free oscillations that can be observed after high-magnitude earthquakes have been studied extensively with gravimeters and low-frequency seismometers over many decades leading to invaluable insight into Earth’s structure. Making use of our detailed understanding of Earth’s normal modes, numerical models are employed for the first time to accurately calculate Earth’s gravitational-wave response, and thereby turn a network of sensors that so far has served to improve our understanding of Earth, into an astrophysical observatory exploring our Universe. In this paper, we constrain the energy density of gravitational waves to values in the range 0.035–0.15 normalized by the critical energy density of the Universe at frequencies between 0.3 and 5 mHz, using ten years of data from the gravimeter network of the Global Geodynamics Project that continuously monitors Earth’s oscillations. This work is the first step towards a systematic investigation of the sensitivity of gravimeter networks to gravitational waves. Further advances in gravimeter technology could improve sensitivity of these networks and possibly lead to gravitational-wave detection.

Figure 1

Current upper limits on GW energy density. These limits were set by pulsar timing observations [23], Doppler-tracking measurements of the Cassini spacecraft [10], monitoring Earth’s free-surface response with seismometers (“Seismic”) [18], and correlating data from the first-generation, large-scale GW detectors LIGO [24]. The new limits resulting from normal-mode measurements are shown as crosses.

Figure 2

Simulated spectrum of spheroidal normal modes around ${}_6{S}_2$. The curves are sums of harmonic oscillator response functions (solid: all spheroidal modes, dashed: all spheroidal quadrupole modes). The values of the red markers correspond to the modes’ $Q$ values.

Figure 3

Earth quadrupole oscillation. The red and blue shapes correspond to the maxima of a quadrupole oscillation separated by half an oscillation period. The green balls mark locations of some of the gravimeters of the GGP network. Here the oscillation is induced by a GW propagating along the north-south axis.

Figure 4

Medians of gravimeter spectra measured in 2012. All gravimeters used in this study show a comparable level of stationary background noise represented by their spectral medians, except for the four gravimeters highlighted in the plot.

Figure 5

Coherence of signals from two levitated spheres in the same gravimeter at Wettzell, Germany. The result is shown as a function of percentile of gravimeter noise excluded from the coherence measurement. A percentile of 90 means that 10% of the loudest spectra were excluded from the coherence measurement. Only the high-$Q$ radial normal mode ${}_0{S}_0$ at about 0.81 mHz contributes significantly to coherence for all times.

Figure 6

Point estimates of GW energy density and errors. Of the 14 original modes, only 12 are plotted here since two pairs, $n=8$, 9, and $n=10$, 11, have been merged to one value each since the chosen frequency resolution cannot resolve them.