Spectrum of gravitational waves from long-lasting primordial sources

We discuss long-lasting gravitational wave sources arising and operating during radiation-dominated stage. Under a set of assumptions, we establish the correspondence between cosmological evolution of a source and the resulting gravitational wave spectrum. Namely, for the source energy density $\rho_s$ falling as a power law characterized by the exponent $\beta$, i.e., $\rho_s \propto 1/a^{\beta}$, where $a$ is the Universe scale factor, the spectrum takes the form $\Omega_{gw} \propto f^{2\beta-8}$ in certain ranges of values of constant $\beta$ and frequencies $f$. In particular, matching to the best fit power law shape of stochastic gravitational wave background discovered recently by Pulsar Timing Array collaborations, one identifies $\beta \approx 5$. We demonstrate the correspondence with concrete examples of long-lasting sources: domain walls and cosmic strings.