Measuring the primordial curvature perturbations from the scalar induced gravitational waves

Scalar induced gravitational waves are produced from primordial curvature perturbations in the second order of perturbations. We constrain the fractional energy density of scalar induced gravitational waves from gravitational wave observations. If there is no detection of scalar induced gravitational waves, the fractional energy density of scalar induced gravitational waves is constrained by some upper limits. Depending on these upper limits, we can obtain the constraints on the power spectrum of the primordial curvature perturbations. For a power-law scalar power spectrum, the constraints on the power spectrum are affected by adding the upper limit of scalar induced gravitational waves from the Square Kilometer Array (SKA). In the standard model, the mean values of the scalar amplitude and the spectral index shift to lower values when the SKA is added to the combination of cosmic microwave background (CMB) and baryon acoustic oscillation (BAO) datasets, namely ln (1010As) = 3.038 +/- 0.013 and Its = 0.9589+0.0021 -0.0011 at 68% confidence level. We also consider the effects of the existing ground-based gravitational-wave detectors, the existing pulsar timing arrays and Five-hundred-meter Aperture Spherical radio Telescope, while the constraints from CMB + BAO datasets are totally within their upper limits of scalar induced gravitational waves. Furthermore, we characterize the scalar fluctuation spectrum in terms of the spectral index Its and its first two derivatives. We calculate corresponding power spectrum of scalar induced gravitational waves theoretically and give the constraints on the running of the spectral index and the running of the running of the spectral index.