Signature of inflation in the stochastic gravitational wave background generated by cosmic string networks

A cosmic string network created during an inflationary stage in the early Universe—defined here as an i-string network—is expected to enter a transient stretching regime during inflation, in which its characteristic length is stretched to scales much larger than the Hubble radius, before attaining a standard evolution once the network reenters the Hubble volume after inflation. During the stretching regime, the production of cosmic string loops and the consequent emission of gravitational radiation are significantly suppressed. Here, we compute the power spectrum of the stochastic gravitational wave background generated by i-string networks using the velocity-dependent one-scale model to describe the network dynamics, and we demonstrate that this regime introduces a high-frequency signature on an otherwise standard spectrum of the stochastic gravitational wave background generated by cosmic strings. We argue that, if observed by current or forthcoming experiments, this signature would provide strong evidence for i-strings and, therefore, for (primordial) inflation. We also develop a simple single-parameter algorithm for the computation of the stochastic gravitational wave background generated by i-strings from that of a standard cosmic string network, which may be useful in the determination of the observational constraints to be obtained by current and forthcoming gravitational wave experiments.

Figure 1

Evolution of $LH$ (a), $L/a$ (b) and $\overline{v}$ (c) with the scale factor $a$ for networks of cosmic strings with different entry times. The case labeled as standard represents a network for which $LH$ and $\overline{v}$ are set to their radiation-era scaling values as the initial conditions. The cosmological parameters used were $h=0.679$, ${\mathrm{\Omega }}_{\mathrm{\Lambda }}^0=0.694$ and ${\mathrm{\Omega }}_r^0{h}^2=2.47\times {10}^{-5}$, in accordance with the Planck data [35]. The value of the scale factor at the present time is taken to be 1, $a_0=1$.

Figure 2

The SGWB spectra, ${\mathrm{\Omega }}_{\mathrm{GW}}{h}^2(f)$, produced by a network of i-strings with $a_e={10}^{-16}$ (orange dash-dotted line) and by a network of strings created at $a_c={10}^{-16}$ (blue solid line). The standard spectrum is depicted in the dashed line. The spectra were calculated with $\alpha ={10}^{-9}$ and $G\mu ={10}^{-7}$.

Figure 3

The SGWB spectra, ${\mathrm{\Omega }}_{\mathrm{GW}}{h}^2(f)$, produced by networks with different entry times (the same as in Fig. 1). The spectra were calculated with $\alpha =0.1$ and $G\mu ={10}^{-7}$.

Figure 4

The SGWB spectra, ${\mathrm{\Omega }}_{\mathrm{GW}}{h}^2(f)$, produced by networks with different values of $\alpha$. In the top panel, (a) cases where loops are considered large are shown, while in the bottom panel, (b) cases in the small-loop regime are plotted. The entry time is $a_e={10}^{-10}$ and the spectra were calculated with $G\mu ={10}^{-8}$.

Figure 5

The SGWB spectra, ${\mathrm{\Omega }}_{\mathrm{GW}}{h}^2(f)$, produced by networks with different values of $G\mu$. The entry time is $a_e={10}^{-10}$ and the spectra were calculated with $\alpha ={10}^{-8}$.

Figure 6

Approximation (solid line) of the SGWB spectrum, ${\mathrm{\Omega }}_{\mathrm{GW}}{h}^2(f)$, following the recipe developed in this section. The dash-dotted line represents the SGWB spectrum produced by a network of i-strings, with $a_e=10^{-5}$ and the dashed line represents the standard SGWB spectrum. For low frequencies, these 3 spectra coincide. The vertical dashed line indicates the position of $f_{\text{cut}}$. The loop size parameter was set to $\alpha =0.1$ and $G\mu ={10}^{-7}$.

Figure 7

Approximation (solid line) of the SGWB spectrum, ${\mathrm{\Omega }}_{\mathrm{GW}}{h}^2(f)$, following the recipe developed in this section. The dash-dotted line represents the SGWB spectrum produced by a network of i-strings, with $a_e={10}^{-15}$ and the dashed line represents the standard SGWB spectrum. The loop size parameter was set to $\alpha ={10}^{-9}$ and $G\mu ={10}^{-7}$.

Figure 8

Approximation (solid line) of the SGWB spectrum, ${\mathrm{\Omega }}_{\mathrm{GW}}{h}^2(f)$, following the recipe developed in this section. The dash-dotted line represents the SGWB spectrum produced by a network of i-strings, with $a_e={10}^{-15}$ and the dashed line represents the standard SGWB spectrum. The loop size parameter was set to $\alpha =2\times {10}^{-5}$ and $G\mu ={10}^{-7}$.

Figure 9

Approximation (solid line) of the SGWB spectrum, ${\mathrm{\Omega }}_{\mathrm{GW}}{h}^2(f)$, following the recipe developed in this section. The dash-dotted line represents the SGWB spectrum produced by a network of i-strings, with $a_e=10^{-5}$ and the dashed line represents the standard SGWB spectrum. The loop size parameter was set to $\alpha =0.1$ and $G\mu ={10}^{-7}$.