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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 $Ω_{\rm{gw}}h^{2}$, via analyzing \emph{Planck} \& ACT \& SPT CMB and DESI BAO data combination. In the standard cosmological model, we establish 95\% CL upper limits of $Ω_{\rm{gw}}h^{2} < 1.0 \times 10^{-6}$ for adiabatic initial conditions and $Ω_{\rm{gw}}h^{2} < 2.7 \times 10^{-7}$ for homogeneous initial conditions, assuming a uniform prior for $Ω_{\rm gw}h^{2}$. In light of dynamical dark energy, we get $Ω_{\rm{gw}}h^{2} < 7.2 \times 10^{-7}$ (adiabatic) and $Ω_{\rm{gw}}h^{2} < 2.4 \times 10^{-7}$ (homogeneous). In contrast, if a log-uniform prior was assumed for $Ω_{\rm gw}h^{2}$, these constraints can become tighter by a factor of $\sim4$, 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 $σ= 2.5 \times 10^{-7}$ (adiabatic) and $σ= 1.0 \times 10^{-7}$ (homogeneous) for ${Ω_{\rm{gw}}h^{2}}$. 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.