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We derived here the factor $\Upsilon$, which quantifies how the gravitational wave spectrum generated by sound waves in the radiation sector grows over time, in a universe with a generic expanding rate set by another dominant energy content. When the dominant energy density satisfies $\rho \propto a^{-3(1+w)}$, we found that $\Upsilon$ has a compact analytical expression: $\Upsilon =\frac{2[1-y^{3(w-1)/2}]}{3(1-w)}$, where $y = a(t)/a(t_s)$ which is the ratio of the scale factor at a later time $t$ to that at $t_s$ when gravitational wave production from sound waves starts. This generic result reduces to that derived previously for radiation-dominated and matter-dominated cases, thus generalizing previous formulas to more general cosmological contexts and providing more accurate results. The derivation relies solely on a stationary source, implying that this generic result of $\Upsilon$ serves as an universal factor in describing the growth of the gravitational wave production and can appear beyond cosmological phase transitions., Comment: 9 pages, 3 figures

We employ tensor network renormalization to explore the dynamical spectra of the easy-axis triangular-lattice antiferromagnet (TLAF) in a magnetic field. Our analysis identifies two distinct low-energy magnon excitations: a gapless Goldstone mode and a gapped mode. At zero field, the spectra display two nearly degenerate roton modes near the M point. With the increase of the magnetic field within the Y-shape superfluid phase, these modes diverge, with the roton excitation vanishing from the Goldstone mode branch, suggesting that the roton dip in this mode may just result from the energy-level repulsion imposed by the roton excitation in the gapped mode. Moreover, the in-plane spectral function shows substantial weight in high energies in the same spin excitation channel where the low-energy roton excitation appears. However, these roton excitations are absent in the V-shape supersolid phase.

The Higgs triplet model (HTM) extends the Standard Model (SM) by one complex triplet scalar (also known as the type-II seesaw model), offering a simple and viable way to account for nonzero neutrino masses. On the other hand, the nontrivial couplings of the triplet to the gauge fields and to the SM Higgs field are expected to influence the topological vacuum structure of the SM, and consequently, the energy and the field configuration of the electroweak sphaleron. The sphaleron process plays a crucial role in dynamically generating the baryon asymmetry of the Universe. In this work, we study the vacuum structure of the gauge and Higgs fields and calculate the saddle-point sphaleron configuration in the HTM. The coupled nonlinear equations of motion of the sphaleron are solved using the spectral method. We find the inclusion of the triplet scalar could in principle significantly change the sphaleron energy compared with the SM. Nevertheless, at zero temperature, the current stringent experimental constraint on the vacuum expectation value of the triplet suppresses the difference. Interestingly, we find that there still exists some narrow parameter space where the sphaleron energy can be enhanced up to $30\%$ compared with the SM case., Comment: 35 pages, 7 figures, 1 table, more discussions and references added, version accepted by JHEP