Stochastic gravitational wave background generated by cosmic string networks: Velocity-dependent one-scale model versus scale-invariant evolution

We compute the power spectrum of the stochastic gravitational wave background generated by cosmic string networks, described by the velocity-dependent one-scale (VOS) model for a wide range of macroscopic and microscopic parameters. The VOS model—which has been shown to provide an accurate macroscopic description of the evolution of cosmic string networks—is used to demonstrate that cosmic string networks are unable to rapidly attain scale-invariant evolution after the transition between the radiation and matter eras. However, in computations of the stochastic gravitational wave background, it is often assumed that the networks experience scale-invariant evolution throughout cosmological history. We demonstrate that this assumption leads to an underestimation of the amplitude and broadness of the peak of the spectrum. As a result, the total energy density of gravitational waves at the present day obtained using the VOS model may be up to 70% larger than that obtained under the linear scaling assumption.

Figure 1

Evolution of $\xi$ (top panel) and of the RMS velocity $\overline{v}$ (lower panel) of a network of cosmic strings with $G\mu ={10}^{-7}$ in a flat $3+1$-dimensional FRW universe, as a function of the scale factor $a$, for various values of $\tilde{c}$ (see label). The cosmological parameters were set to ${\Omega }_{\Lambda 0}=0.728$, ${\Omega }_{r0}{h}^2=2.47\times {10}^{-5}$ and $h=0.704$, which correspond to the WMAP 7-year data combined with the baryon acoustic oscillation data from the Sloan Digital Sky Survey and determinations of $H_0$ using the Hubble Space Telescope [62]. The scale factor at the time of radiation and matter equality is $a_{\mathrm{eq}}$, while its value at the present time was set to unity.

Figure 2

The SGWB spectrum, ${\Omega }_{\mathrm{GW}}{h}^2$, as a function of the frequency, $f$, for various values of $\alpha >\Gamma G\mu$. The solid lines represent the spectra created by a cosmic string network described by the VOS model, while the dashed lines represent the spectra obtained using Model II. Here, we assumed that $G\mu ={10}^{-7}$ and $\tilde{c}=0.23$, and only the fundamental mode of emission was considered ($n_s=1$).

Figure 3

The SGWB spectrum, ${\Omega }_{\mathrm{GW}}{h}^2$, as a function of the frequency, $f$, for various values of $\alpha \le \Gamma G\mu$. The solid lines represent the spectra generated using Model I, while dashed lines represent the spectra obtained using Model II. As in Fig. 2, we have assumed that $G\mu ={10}^{-7}$ and $\tilde{c}=0.23$, and $n_s=1$.

Figure 4

The SGWB spectrum, ${\Omega }_{\mathrm{GW}}{h}^2$, as a function of the frequency, $f$, for various values of $G\mu$. The solid lines represent the spectra generated using Model I, and the dashed lines represent the spectra obtained using Model II. The loop size parameter was set to $\alpha ={10}^{-7}$, the loop-chopping efficiency was set to $\tilde{c}=0.23$, and only the fundamental mode of emission was considered ($n_s=1$).

Figure 5

The SGWB spectrum, ${\Omega }_{\mathrm{GW}}{h}^2$, as a function of the frequency, $f$, for different values of $n_s$. The solid lines represent the spectra generated by a cosmic string network described by VOS model (Model I), while the dashed lines are spectra obtained using Model II. Here, we have considered kinky loops (with $q=2$) with a loop size parameter $\alpha ={10}^{-1}$, and we have assumed that $G\mu ={10}^{-7}$ and $\tilde{c}=0.23$.

Figure 6

The SGWB spectrum, ${\Omega }_{\mathrm{GW}}{h}^2$, as a function of the frequency, $f$, for different values of $n_s$. The solid lines represent the spectra generated using Model I, while the dashed lines are the spectra obtained using Model II. Here, we considered cuspy loops (with $q=4/3$) with $\alpha ={10}^{-1}$, and we have assumed that $G\mu ={10}^{-7}$ and $\tilde{c}=0.23$.

Figure 7

The SGWB spectrum, ${\Omega }_{\mathrm{GW}}{h}^2$, as a function of the frequency, $f$, for different values of $n_s$. The solid lines represent the spectra generated by a cosmic string network described by the VOS model (Model I), while the dashed lines are obtained using Model II. Here, we considered kinky loops (with $q=2$) with a loop size parameter $\alpha ={10}^{-8}$, and we have assumed that $G\mu ={10}^{-7}$ and $\tilde{c}=0.23$.

Figure 8

The SGWB spectrum, ${\Omega }_{\mathrm{GW}}{h}^2$, as a function of the frequency, $f$, for different values of $n_s$. The solid lines represent the spectra generated using Model I, while the dashed lines are obtained using Model II. Here, we considered cuspy loops (with $q=4/3$) whose size is determined by $\alpha ={10}^{-8}$, and we have assumed that $G\mu ={10}^{-7}$ and $\tilde{c}=0.23$.

Figure 9

Some examples of SGWB spectra generated by cosmic string networks (see label for details) that would not seem to be within the detection range of the five-year (pink area) and ten-year (gray area) European Pulsar Timing Array (EPTA) experiments [82], when scale invariance is assumed (dashed lines). In these examples, the equivalent spectra obtained using the VOS model (solid lines) are within the detection range of the EPTA experiments.