New constraints on cosmological gravitational waves from CMB and BAO in light of dynamical dark energy

In this work, we derive upper limits on the physical energy-density fraction today of cosmological gravitational waves, denoted by ${\mathrm{\Omega }}_{\mathrm{g}\mathrm{w}}{h}^2$, via analyzing \emph{Planck} & ACT & SPT CMB and DESI BAO data combination. In the standard cosmological model, we establish 95% CL upper limits of ${\mathrm{\Omega }}_{\mathrm{g}\mathrm{w}}{h}^2<1.0\times {10}^{-6}$ for adiabatic initial conditions and ${\mathrm{\Omega }}_{\mathrm{g}\mathrm{w}}{h}^2<2.7\times {10}^{-7}$ for homogeneous initial conditions, assuming a uniform prior for ${\mathrm{\Omega }}_{\mathrm{g}\mathrm{w}}{h}^2$. In light of dynamical dark energy, we get ${\mathrm{\Omega }}_{\mathrm{g}\mathrm{w}}{h}^2<7.2\times {10}^{-7}$ (adiabatic) and ${\mathrm{\Omega }}_{\mathrm{g}\mathrm{w}}{h}^2<2.4\times {10}^{-7}$ (homogeneous). In contrast, if a log-uniform prior was assumed for ${\mathrm{\Omega }}_{\mathrm{g}\mathrm{w}}{h}^2$, these constraints can become tighter by a factor of $\sim 4$, suggesting the results to be prior-sensitive. Furthermore, we project the sensitivity achievable with LiteBIRD & CMB Stage-IV measurements of CMB and CSST observations of BAO, forecasting 68% CL uncertainties of $\sigma =2.5\times {10}^{-7}$ (adiabatic) and $\sigma =1.0\times {10}^{-7}$ (homogeneous) for ${\mathrm{\Omega }}_{\mathrm{g}\mathrm{w}}{h}^2$. The constraints we obtained in this work provide critical benchmarks for exploring the cosmological origins of gravitational waves within the frequency band $f\gtrsim {10}^{-15}$,Hz and potentially enable joint analysis with direct gravitational-wave detection sensitive to this regime.