Gravitational waves from cosmic strings for pedestrians

Abstract

Cosmic strings represent an attractive source of gravitational waves (GWs) from the early Universe. However, numerical computation of the GW signal from cosmic strings requires the evaluation of complicated integral and sum expressions, which can become computationally costly in large parameter scans. This motivates us to rederive the GW signal from a network of local stable cosmic strings in the Nambu-Goto approximation and based on the velocity-dependent one-scale model from a ``pedestrian'' perspective. That is, we derive purely analytical expressions for the total GW spectrum, which remain exact wherever possible and whose error can be tracked and reduced in a controlled way in crucial situations in which we are forced to introduce approximations. In this way, we obtain powerful formulas that, unlike existing results in the literature, are valid across the entire frequency spectrum and across the entire conceivable range of cosmic-string tensions. We provide an in-depth discussion of the GW spectra thus obtained, including their characteristic break frequencies and approximate power-law behaviors, comment on the effect of changes in the effective number of degrees of freedom during radiation domination, and conclude with a concise summary of our main formulas that can readily be used in future studies.

Figures (16)

• Figure 1: Time evolution of the reduced correlation length $\xi$ (teal) and the RMS velocity $v_\infty$ (deep purple) describing the long-string network. The evolution is obtained by numerically solving the coupled differential equations \ref{['eq:DifferentialequationsVOS']}. The vertical dashed lines indicate matter--radiation equality $t_{\rm eq}$ and cosmological constant--matter equality $t_{\rm eq}'$.
• Figure 2: Fundamental GW spectrum for the RR case, a string tension of $G\mu = 10^{-10}$ and an initial time $t_{\rm ini}=t_{\rm fric}$. Upper panel: Our exact numerical result based on the VOS model accounting for the full time dependence of all relevant quantities and for a fixed effective number of DOFs as explained in Section \ref{['subsec:Numerical']} (teal), our analytical result derived in this paper (deep purple), and the analytical result derived in Ref. Sousa_2020 (light green, dash-dotted). Grey dashed lines show characteristic frequencies of the spectrum, and black dashed lines indicate different power-law behaviors of the spectrum. Lower panel: Relative deviation $\delta_\Omega$ (cf. \ref{['eq:DeltaOmega']}) of the two analytical results from our numerical result.
• Figure 3: Left panel: Very-low-frequency region of the spectra shown in Fig. \ref{['fig:RR1']}. Our numerical spectrum and the analytical spectrum of this work drop to zero as they approach $f_{\rm rr}^{\rm min}$, while the result of Ref. Sousa_2020 are cut off by hand. Right panel: Region of the spectra shown in Fig. \ref{['fig:RR1']} in which the transition from the plateau to the high-frequency regime occurs. The spectrum determined in Ref. Sousa_2020 deviates in this regime by $\sim 20\%$ from our numerically calculated one, while our analytical result shows excellent agreement.
• Figure 4: Fundamental GW spectrum for the RR case, a string tension of $G\mu = 10^{-20}$ and an initial time $t_{\rm ini}=t_{\rm fric}$. Panels and color codes are the same as in Fig. \ref{['fig:RR1']}. The spectrum derived in Ref. Sousa_2020 does not appear in the plot since it turns negative.
• Figure 5: UHF regime of the fundamental GW spectrum for the RR case, string tensions of $G\mu = 10^{-7}$ (left) and $G\mu = 10^{-10}$ (right) and initial times given by $t_{\rm ini} = t_{\rm fric}$. Panels and color codes are the same as in Fig. \ref{['fig:RR1']}. The grey dashed line shows the frequency at which the UHF effects become relevant.