Gravitational Waves and Proton Decay: Complementary Windows into Grand Unified Theories

Proton decay is a smoking gun signature of grand unified theories (GUTs). Searches by Super-Kamiokande have resulted in stringent limits on the GUT symmetry-breaking scale. The large-scale multipurpose neutrino experiments DUNE, Hyper-Kamiokande, and JUNO will either discover proton decay or further push the symmetry-breaking scale above $10^{16}\mathrm{GeV}$. Another possible observational consequence of GUTs is the formation of a cosmic string network produced during the breaking of the GUT to the standard model gauge group. The evolution of such a string network in the expanding Universe produces a stochastic background of gravitational waves which will be tested by a number of gravitational wave detectors over a wide frequency range. We demonstrate the nontrivial complementarity between the observation of proton decay and gravitational waves produced from cosmic strings in determining SO(10) GUT-breaking chains. We show that such observations could exclude SO(10) breaking via flipped $\mathrm{SU}(5)\times \mathrm{U}(1)$ or standard SU(5), while breaking via a Pati-Salam intermediate symmetry, or standard $\mathrm{SU}(5)\times \mathrm{U}(1)$, may be favored if a large separation of energy scales associated with proton decay and cosmic strings is indicated. We note that recent results by the NANOGrav experiment have been interpreted as evidence for cosmic strings at a scale of $\sim {10}^{14}\mathrm{GeV}$. This would strongly point toward the existence of GUTs, with SO(10) being the prime candidate. We show that the combination with already available constraints from proton decay allows us to identify preferred symmetry-breaking routes to the standard model.

Figure 1

The breaking chains of SO(10) to $G_{\mathrm{SM}}$ are shown along with their terrestrial and cosmological signatures, where $G_x$ represents either $G_{3221}$ or $G_{421}$. Defects with only cosmic strings (including cosmic strings generated from preserved discrete symmetries) are denoted as blue solid arrows. Those including unwanted topological defects (monopoles or domain walls) are indicated by red dotted arrows. The instability of embedded strings is not considered. Removing an intermediate symmetry may change the type of unwanted topological defect but will not eliminate them. The highest possible scale of inflation, which removes unwanted defects, is assumed in this diagram.

Figure 2

SGWB predicted from undiluted (solid black lines) and diluted (dashed blue lines) cosmic string networks, where ${\mathrm{\Lambda }}_{\mathrm{cs}}=10^{10,11,\dots ,15}\mathrm{GeV}$ are input. $\tilde{z}$ denotes the redshift when strings return to the horizon, namely, $H(\tilde{z})L(\tilde{z})=1$. Current (hatched) and future (colored) experimental limits are shown as a comparison.

Figure 3

GUTs constrained by observations of GWs and proton decays. Left: Current (hatched) and future (solid) exclusion limits of energy scales of cosmic string formation, proton decays, and inflation. ${\mathrm{\Lambda }}_{\mathrm{pd}}\sim {\mathrm{\Lambda }}_1$ is approximated, and the exclusion limit of ${\mathrm{\Lambda }}_{\mathrm{cs}}$ is shown in the undiluted case only. Right: Potential conclusions of GUT properties based on observations of GWs and proton decays in next-generation experiments.