Measuring the speed of scalar induced gravitational waves from observations

We investigate the scalar induced gravitational waves which propagate with a speed different from the speed of light. First, we analytically calculate the expression of the power spectrum of the scalar induced gravitational waves which is based on the speed and the spectrum of the primordial curvature perturbations. Then, we discuss several scalar power spectra and obtain corresponding fractional energy density, such as the monochromatic power spectrum, the scale invariant power spectrum and the power-law power spectrum. Finally, we constrain the scalar induced gravitational waves and evaluate the signatures of the speed from the combination of CMB+BAO and gravitational waves observations. The numerical results are obvious to reveal the influence of speed of scalar induced gravitational waves.