We consider the scalar induced gravitational waves from the cosmic microwave background (CMB) observations and the gravitational wave observations. In the \(\Lambda \)CDM+r model, we constrain the cosmological parameters within the evolution of the scalar induced gravitational waves by the additional scalar power spectrum. The two special cases called narrow power spectrum and wide power spectrum have influence on the cosmological parameters, especially the combinations of Planck18+BAO+BK15+LISA. We also compare these numerical results from four datasets within LIGO, LISA, IPTA and FAST projects, respectively. The constraints from FAST have a significant impact on tensor-to-scalar ratio. Besides, we only consider the relic density of induced gravitational waves with respect to different frequencies from CMB scale to high frequencies including the range of LIGO and LISA.

我们考虑了来自宇宙微波背景 (CMB) 观测和引力波观测的标量诱导引力波。在\(\Lambda \) CDM+ r在模型中，我们通过额外的标量功率谱来约束标量诱导引力波演化中的宇宙学参数。窄功率谱和宽功率谱这两种特殊情况对宇宙学参数有影响，尤其是Planck18+BAO+BK15+LISA的组合。我们还分别比较了 LIGO、LISA、IPTA 和 FAST 项目中四个数据集的这些数值结果。FAST 的约束对张量与标量比有重大影响。此外，我们只考虑了从 CMB 尺度到包括 LIGO 和 LISA 范围在内的高频不同频率的诱导引力波的残余密度。