Gravitational wave bursts from cosmic string cusps and pseudocusps

We study the relative contribution of cusps and pseudocusps, on cosmic (super)strings, to the emitted bursts of gravitational waves. The gravitational wave emission in the vicinity of highly relativistic points on the string follows, for a high enough frequency, a logarithmic decrease. The slope has been analytically found to be $-4/3$ for points reaching exactly the speed of light in the limit $c=1$. We investigate the variations of this high-frequency behavior with respect to the velocity of the points considered, for strings formed through a numerical simulation, and we then compute numerically the gravitational waves emitted. We find that for string points moving with velocities as far as ${10}^{-3}$ from the theoretical (relativistic) limit $c=1$, gravitational wave emission follows a behavior consistent with that of cusps, effectively increasing the number of cusps on a string. Indeed, depending on the velocity threshold chosen for such behavior, we show that the emitting part of the string worldsheet is enhanced by a factor $\mathcal{O}(10^3)$ with respect to the emission of cusps only.

Figure 1

Schematic representation of the ${\dot{\mathbf{X}}}_{+}$ (in red) and ${\dot{\mathbf{X}}}_{-}$ (in blue) curves on the unit sphere, with black points highlighting the cusp configurations, that is, the crossings of these curves.

Figure 2

Visualization of string configuration used in our numerical simulations depicting a light string stretched between two junctions with heavier fixed strings.

Figure 3

String worldsheets using space and time coordinates ($\mathrm{\Sigma }$ in red) and null coordinates ($\overline{\mathrm{\Sigma }}$ in blue). Even though $\overline{\mathrm{\Sigma }}$ is twice as large as $\mathrm{\Sigma }$, both contain the same information. Indeed, ${\mathrm{\Sigma }}_{\mathrm{A}}={\overline{\mathrm{\Sigma }}}_{\mathrm{NW}}$ and ${\overline{\mathrm{\Sigma }}}_{\mathrm{SW}}$ are the symmetric of ${\overline{\mathrm{\Sigma }}}_{\mathrm{NE}}$ and ${\overline{\mathrm{\Sigma }}}_{\mathrm{SE}}$ (respectively), under the symmetry $\sigma \to L-\sigma$; also, the symmetric of ${\mathrm{\Sigma }}_{\mathrm{B}}$ (using again $\sigma \to L-\sigma$) is equivalent to ${\overline{\mathrm{\Sigma }}}_{\mathrm{SW}}$ due to the $L$ periodicity of $\overline{\mathrm{\Sigma }}$.

Figure 4

Calculated slope for a single cusp using a different number of frequency points in the frequency interval [2.52,1000] Hz defined in Sec. 3. We find that the slope converges quickly enough that significantly optimizing computational time with a lower number of points is an acceptable approximation, allowing for an analysis of a much larger sample of points.

Figure 5

Calculated slope of the high-frequency end of the GW waveform’s power spectrum with respect to $\log (1-v)$, the logarithmic deviation of velocity from $c=1$, using all of the 10 326 points in the velocity range $1>v\ge 0.99$. The convoluted moving point average is represented by the dot-dashed horizontal black line along with a $\pm \sigma$ deviation band in red. The velocity ranges are split into four regions, namely cusps [in red, for $\log (1-v)\in [-10,-6]$], pseudocusps (in orange, for $[-6,-3]$), external (in grey, for $[-3,-2]$) and transitional (in green, for $[-3.41,-2.53]$). The vertical dashed lines limit the transition region, giving a conservative and a relaxed threshold.

Figure 6

Zoomed-in region of Fig. 5, where the high-frequency behavior transitions from its $-4/3$ analytical slope, away from the cusp behavior.

Figure 7

(a), (b) and (c): Relationship plots for cusps and pseudocusps per string over the total of 119 strings highlighting the approximate $2:1$ relationship of cusps to pseudocusps as argued in Ref. [38]. (d): The number of events contributing to each velocity region.

Figure 8

Representations of the close approach of the ${\dot{\mathbf{X}}}_{+}$ and ${\dot{\mathbf{X}}}_{-}$ curves on the unit sphere, used to identify cusps (direct crossings) and pseudocusps (near crossings) respectively.

Figure 9

Simulation of a cusp’s formation, with snapshots of a section of the string at different instants both before and after the formation of the cusp. The dotted line represents the time evolution of the point $\sigma =x$ on the string which becomes a cusp at the time $\tau =y$.

Figure 10

High-frequency window of the GW power spectrum for all the cusps (20) of a random string, as well as the linear regression yielding the calculated slope, in red. Only 26 frequency points in the frequency range [2.52,1000] Hz have been used.

Figure 11

Calculated slope of the GW power spectrum with respect to $\log (1-v)$, the logarithmic velocity, in the neighborhood of a cusp. The velocity ranges (vertical and horizontal dashed lines) are defined as in Fig. 5. The red dot represents the cusp event, while the red (black) curve shows the approach to (respectively, the decline out of) the cusp, that is $\sigma <{\sigma }^{(\mathrm{c})}$ or $\tau <t^{(\mathrm{c})}$ (respectively $\sigma >{\sigma }^{(\mathrm{c})}$ or $\tau >t^{(\mathrm{c})}$). Panel (a) shows the behavior of the vicinity of the cusp in the spatial $\sigma$ direction, while panel (b) shows the same in the temporal $\tau$ direction.