Constraints on Dark Energy from the Gravitational Wave Background

Current observational data indicate that dark energy (DE) is a cosmological constant without considering its conclusiveness evidence. Considering the dynamic nature of $\Lambda$ individually as a function of time and the scale factor, we review their effects on the gravitational waves. This article is a continuation of the previous work \textit{JHEAp 36 (2022) 48-54}, in which DE only was based on Hubble's parameter and/or its derivatives. For the DE model based on the scale factor ($a^{-m}$), the results showed that the parameter $m$ is more limited as $2 < m \leqslant 3$ compared with the other models and due to the small value of DE density at the early universe. It is only in the mode $m=3$ that DE affects the low-frequency gravitational waves when its frequency is less than the $10^{-3}$Hz in a matter-dominated epoch. The broad bound on reducing the amplitude and the "B-B" polarization multipole coefficients, from maximum to minimum, is for the models developed based on the Hubble parameter function. There are primary sources of low- and very low-frequency GWs, such as the coalescence of massive black hole binaries with $M_{bh} > 10^{3} M_{sun}$, to determine the type of DE by mHz frequency space experiments (e.g., LISA) and by nHz-range NANOGrav 15-year data.