Gravitational Wave Signals in a Promising Realization of SO(10) Unification

Abstract

We investigate gravitational wave signals in a non-supersymmetric grand unified model where the group $SO(10)$ is broken in two steps to the Standard Model gauge group. We calculate the analytical form of the one-loop effective potential responsible for the first step of symmetry breaking and show that it can lead to a first-order phase transition with gravitational wave production. We also determine the gravitational wave background produced by the primordial plasma of relativistic particles. The present experimental sensitivity is still far from the expected signals, but could be in reach of novel detector concepts.

Figures (10)

• Figure 1: Possible scenarios for the completion of the phase transition. In scenario "A", the phase transition happens at the scale of grand unification and after the inflation scale, so its effects are visible today. Scenario "B" refers to the case that the phase transition does not complete at $M_{\text{GUT}}$ but is completed after inflation. We briefly mention this possibility in Section \ref{['sec:incompPT']}. Here $M_{\text{Pl}}$, $M_{\rm{Inf }}$, $M_{\text{GUT}}$, $M_{\rm{PT}}$, $M_{R}$ and $M_{\text{EW}}$ are respectively the Planck mass, the inflation scale, the FOPT scale, the GUT scale, the scale at which $SU(2)_R$ is broken, and finally the EW scale.
• Figure 2: The left panel represents the running of the Minimal $\mathbf{G}_{3221}$ model, $\mathbf{G}^{\rm M}_{3221}$ (containing only $\mathbf{45}\oplus\mathbf{126}\oplus \mathbf{10}$), with $M_{R}=(2.9\pm 1.0) \times 10^{9}\,\text{GeV}$ and $M_{\text{GUT}}= (1.60\pm 1.0) \times 10^{16}\,\text{GeV}$. The gauge couplings of the groups of the SM are run up to $M_{R}$ and those of $SU(3)_C\times SU(2)_L \times SU(2)_R\times U(1)_{B-L}$ from $M_{R}$ and $M_{\text{GUT}}$. The vertical gray band around $2.8\times 10^9\,\text{GeV}$ represents the band allowed for $M_{R}$ for this model and the model of the right plot. The right plot represents the running of the $\mathbf{G}_{3221}^\text{M}$ model plus a fermionic $SU(2)_R$-$SU(2)_L$ bi-doublet Dark Matter candidate (that of the third panel Fig. 6 of Biswas:2022cyh). We present this last plot as an example that the addition of matter can alter the running (see Eq. \ref{['eq:betagigen']} and the text below) of the gauge coupling constants and reduce the unification scale.
• Figure 3: Phase transition parameters ($\alpha$, $\beta/H_*$) and GW density as a function of $a_0$ for the scales $M_{\text{GUT}}=1.6 \times 10^{16}\,$GeV, left, and $M_{\text{GUT}}=1.0 \times 10^{15}\,$GeV, right. The (red) dots represent the benchmark points listed in Tables \ref{['tbl:BP_v1016GeV']} and \ref{['tbl:BP_v1015GeV']}.
• Figure 4: GW parameter space for $a_0$ vs. $a_2$, where $a_2$ is restricted to the region where the spectrum does not contain tachyons. The four points marked by stars are our benchmark points. The yellow region has no FOPT or is numerically difficult to calculate. In the hatched region, $\Delta^\text{2-loop}>0.5$. We denote the quadratic term in the scalar potential obtained from the tree-level condition \ref{['eq:mu2a0a2tree']} by $\mu_0^2$. In contrast, $\mu^2$ is the corrected value of this parameter ensuring that the $1$-loop-corrected scalar potential has a minimum at $v$.
• Figure 5: The same as in Fig. \ref{['fig:cvgce_of_min1loop_1_1p6e16']} but for $M_{\text{GUT}}=1.0\times 10^{15}\,\text{GeV}$.