Probing the scalar-induced gravitational waves with the Five-hundred-meter Aperture Spherical radio Telescope and the Square Kilometer Array

Gravitational wave astronomy presents a promising opportunity to directly observe scalar-induced gravitational waves originating from the early universe. Experiments, including ground-based interferometers like LIGO and Virgo, and the Pulsar Timing Array, such as FAST and SKA, are poised to significantly enhance sensitivity to these gravitational waves. In this paper, we combined Cosmic Microwave Background and Baryon Acoustic Oscillation datasets with upper or lower limits of the stochastic gravitational wave background provided by FAST or SKA, to constrain scalar-induced gravitational waves. To provide a comprehensive forecast, we consider two scenarios at a frequency: one where FAST or SKA does not detect scalar-induced gravitational waves, thereby setting an upper limit on the fractional energy density; and another where these waves are detected successfully, thus establishing a lower limit. In the $\mathrm{\Lambda }$CDM+$r$ model, the scalar spectral index of the power-law power spectrum is constrained to $n_s=0.9598_{-0.0009}^{+0.0013}$ from the combinations of CMB+BAO+SKA datasets in the upper limit scenario where scalar-induced gravitational waves propagate at the speed of light. The constraint shifts to $n_s=0.9697\pm 0.0033$ in the lower limit scenario. Comparing with the constraint from the combinations of CMB+BAO datasets, the scalar spectral index $n_s$ in the upper limit scenario exhibits significant changes, which could serve as an indicator for detecting scalar-induced gravitational waves. In the $\mathrm{\Lambda }$CDM+${\alpha }_s$+$r$ model and $\mathrm{\Lambda }$CDM+${\alpha }_s$+${\beta }_s$+$r$ model, the running of the scalar spectral index ${\alpha }_s$ and the running of the running ${\beta }_s$ also show notable variations, suggesting potential indicators. The numerical findings clearly demonstrate the impact of the upper and lower limits provided by FAST or SKA.