Anisotropies in the gravitational wave background as a probe of the cosmic string network

Pulsar timing arrays are powerful tools to test the existence of cosmic strings by searching for the gravitational wave (GW) background. The amplitude of the background connects to information on cosmic strings such as the tension and string network properties. In addition, one may be able to extract more information on the properties of cosmic strings by measuring anisotropies in the GW background. In this paper, we provide estimates of the level of anisotropy expected in the GW background generated by cusps on cosmic strings. We find that the anisotropy level strongly depends on the initial loop size $\alpha$, and thus we may be able to put constraints on $\alpha$ by measuring the anisotropy of the GW background. We also find that certain regions of the parameter space can be probed by shifting the observation frequency of GWs.

Figure 1

The GW spectra originating from cusps on cosmic string loops are plotted in terms of ${\mathrm{\Omega }}_{\mathrm{GW}}{h}^2$ by changing the values of the initial loop size $\alpha$. The cosmic string tension is fixed at $G\mu ={10}^{-11}$, and the reconnection probability is $p=1$. The sensitivity of the SKA is shown by the black dotted line. The shaded areas represent the 68% (dark gray) and 99.7% (light gray) confidence intervals of the theoretically predicted GW amplitude from SMBH binaries according to Ref. [54].

Figure 2

Parameter space accessible by SKA (light blue) for $p=1$ and $p={10}^{-3}$. The blue region is the space which is already excluded by the current pulsar constraints. The black dotted lines correspond to contour lines of ${\mathrm{\Omega }}_{\mathrm{GW}}{h}^2={10}^{-13}$, ${10}^{-12}$, ${10}^{-11}$, ${10}^{-10}$, and $2.2\times {10}^{-10}$ from bottom to top.

Figure 3

The expected number of GW bursts per logarithmic strain, $dR/d\ln \tilde{h}$, for the fixed frequency $f=1/10\mathrm{years}=3.17\times {10}^{-9}\mathrm{Hz}$. From the left to the right, we show parameter dependencies of $G\mu$, $\alpha$, and $p$. The vertical axis is expected numbers of GW bursts per 10 years.

Figure 4

The contribution to the integral of Eq. (15) for each logarithmic strain bin with the same parameter set as in Fig. 3.

Figure 5

The rate shown in terms of the redshift with the same parameter set as in Fig. 3. We fix the strain amplitude to be the one which gives the biggest contribution to the value of ${\mathrm{\Omega }}_{\mathrm{GW}}$ (the value of $\tilde{h}$ at the local maximum in Fig. 4).

Figure 6

The anisotropy power $C_{\ell }/C_0$ for different values of the initial loop size: $\alpha =1.45\times {10}^{-9}$ (black), $1.5\times {10}^{-9}$ (red), and ${10}^{-4}$ (blue). We set $G\mu ={10}^{-11}$ and $p=1$ and assume observation at $f=1/10\mathrm{year}=3.17\times {10}^{-9}\mathrm{Hz}$.

Figure 7

The anisotropy power $C_{\ell }/C_0$ of the dipole moment shown as a function of $\alpha$. We assume $G\mu ={10}^{-11}$ and $p=1$. The different colors describe different observation frequencies; $f=1/10\mathrm{year}=3.17\times {10}^{-9}\mathrm{Hz}$ (black), $3.17\times {10}^{-8}\mathrm{Hz}$ (red), $3.17\times {10}^{-7}\mathrm{Hz}$ (blue), and $3.17\times {10}^{-6}\mathrm{Hz}$ (green).

Figure 8

The anisotropy power $C_{\ell }/C_0$ of the dipole moment, shown as a function of $\alpha$. The observation frequency is assumed to be $f=1/10\mathrm{year}=3.17\times {10}^{-9}\mathrm{Hz}$. In the left panel, we change the value of the tension $G\mu$ by fixing $p=1$; $G\mu ={10}^{-10}$ (red), ${10}^{-11}$ (black), and ${10}^{-12}$ (green). In the right panel, we change the value of the reconnection probability by fixing $G\mu ={10}^{-11}$; $p=1$ (black), ${10}^{-1}$ (red), ${10}^{-2}$ (blue), and ${10}^{-3}$ (green).

Figure 9

The anisotropy power $C_{\ell }/C_0$ of the dipole moment shown as a function of the observation frequency. The different colors describe different values of the initial loop size: $\alpha ={10}^{-9}$ (black), ${10}^{-10}$ (red), ${10}^{-11}$ (blue), and ${10}^{-12}$ (green). The other parameters are set as $G\mu ={10}^{-11}$ and $p=1$.