Gravitational wave echo of relaxion trapping

To solve the hierarchy problem, the relaxion must remain trapped in the correct minimum, even if the electroweak symmetry is restored after reheating. In this scenario, the relaxion starts rolling again until the backreaction potential, with its set of local minima, reappears. Depending on the time of barrier reappearance, Hubble friction alone may be insufficient to retrap the relaxion in a large portion of the parameter space. Thus, an additional source of friction is required, which might be provided by coupling to a dark photon. The dark photon experiences a tachyonic instability as the relaxion rolls, which slows down the relaxion by backreacting to its motion, and efficiently creates anisotropies in the dark photon energy-momentum tensor, sourcing gravitational waves. We calculate the spectrum of the resulting gravitational wave background from this new mechanism and evaluate its observability by current and future experiments. We further investigate the possibility that the coherently oscillating relaxion constitutes dark matter and present the corresponding constraints from gravitational waves.

Figure 1

Available parameter space for $\xi =100$ (top), and for $\xi =10$ (bottom). The red and orange shaded regions are excluded by the indicated constraints or combinations thereof. Above the red solid line, the relaxion decay constant becomes super-Planckian. The gray dashed line encloses the parameter space in which relaxation can be realized without dark photon friction, which is discussed in more details in Appendix pp1. The prospective GW sensitivity of $\mu \text{Ares}$ (green) as well as SKA after an observation period of 5 years (turquoise) and 20 years (blue) is indicated by the respective colored regions. In the purple colored region, a subrange of the viable reappearance temperatures can be excluded using current NANOGrav data from the 11 year dataset. The regions bounded by the colored dotted lines need super-Planckian decay constants to be probed by the respective experiment. In the lower panel, the gray shaded region can reproduce our best-fit spectrum (at $T_{\mathrm{ra}}\sim 20\mathrm{MeV}$) to the potential stochastic GW background seen in the recent NANOGrav data. An animated version of the lower panel of the figure, explicitly showing the dependence on the reappearance temperature, can be found in the Supplemental Material [20] of this paper.

Figure 2

Allowed range of reappearance temperatures $T_{\mathrm{ra}}$ as a function of the cutoff scale $\mathrm{\Lambda }$, while fixing $f_{\varphi }$ to the value reproducing the measured DM abundance, Eq. (16), assuming $T_{\mathrm{osc}}=T_{\mathrm{ra}}$. The blue shaded region is excluded since ${\mathrm{\Lambda }}_{\mathrm{br}}>v_H$, while in the green shaded regions, the CMB bound on $\mathrm{\Delta }{N}_{\mathrm{eff}}$, Eq. (25), is violated. In the orange shaded region, the dark photon energy density further dominates the Universe at reappearance. The dashed lines are contours of constant $m_{\varphi }{f}_{\varphi }/T_{\mathrm{ra}}^2$, which sets the amplitude of the GW spectrum; cf. Eq. (2).

Figure 3

Available parameter space (black framed region) for relaxion DM in the relaxion mass $m_{\varphi }$ vs mixing angle $\sin {\theta }_{h\varphi }$ plane. The red and orange shaded regions are excluded by the indicated constraints of combinations thereof. The colored regions inside the viable DM space can be probed via GWs in $\mu \text{Ares}$ (green) or SKA (blue/turquoise). The light shading and solid lines indicate points that can be probed for a subrange of reappearance temperatures, whereas the darker shaded parts enclosed by dotted lines are accessible for all valid $T_{\mathrm{ra}}$. An animated version of the plot scanning the reappearance temperature is enclosed in the Supplemental Material [20] of this work.

Figure 4

GW spectra for the parameter points listed in Table 1. The orange spectrum labeled “$*$” corresponds to our best-fit to the NANOGrav 12.5 year dataset. The colored regions depict the projected power-law integrated sensitivity curves of LISA (red), $\mu \text{Ares}$ (green), and SKA (turquoise and blue, assuming 5 and 20 years of observation, respectively) and the current exclusion from the NANOGrav 11 year dataset (purple).

Figure 5

Values of the peak frequency and amplitude of the GW spectrum which can be obtained in the relaxion DM scenario. The edges of the polygon correspond to the minimal and maximal amplitudes which can be obtained for $\xi =100$ (solid lines) and $\xi =10$ (dashed lines), considering the case when the relaxion starts to oscillate immediately after barrier reappearance.

Figure 6

Best-fit point (cross) to the NANOGrav 12.5 year data as well as $1\sigma$ (dark) and $2\sigma$ (light) regions, fixing $\xi$ to 10 (blue), 25 (orange), and 100 (green). The dotted lines enclose the region that can be obtained without being in tension with $\mathrm{\Delta }{N}_{\mathrm{eff}}$ or BBN at the respective value of $\xi$.

Figure 7

Available parameter space in the minimal relaxion scenario when the reheating temperature is higher than the EW scale. The red and orange shaded regions are excluded by the indicated constraints or combinations thereof. The blue shaded region is further excluded if the relaxion is not coupled to the dark photon source term, leading to a field excursion of more than $2\delta$. The dark photon production cannot occur in the region above the gray dashed line. Relaxion DM can be realized within the solid and dashed black contours, respectively, depending on whether a coupling to dark photons is assumed or not. In the turquoise parameter region, the condition $\mathrm{\Delta }\theta \lesssim 2\delta$ further leads to an overproduction of relaxion DM in the minimal scenario.

Figure 8

Simulated (colored lines) and expected (dashed black line) amplitude of the dark photon modes at different times. The deep (light) colored lines correspond to the positive (negative) helicity. At each time, the expected peak momentum is indicated by the vertical dashed line in the corresponding color, whereas the black-dashed vertical line indicates the upper bound $k<k_{\mathrm{pp}}$ on the tachyonic dark photon momentum.

Figure 9

Spectral shape of the stochastic GW background for $\xi =10$ (blue), 30 (orange), and 100 (green). The solid lines correspond to the numeric calculation, whereas the dashed ones are the analytic approximation. The dotted lines indicate the IR and UV behavior, and the black cross marks our estimate of the peak amplitude.

Figure 10

Simulation (full colors) and analytic approximation (light colors) of the GW spectrum for a reappearance temperature of $T_{\mathrm{ra}}=150\mathrm{GeV}$ (blue) and 750 GeV (orange). The vertical dashed lines indicate the expected peak frequency, whereas the dot-dashed and dotted curves correspond to the positive and negative helicity contributions to the simulated spectrum.