Gravitational-Wave Bursts from Cosmic String Cusps
Cosmic strings are hypothetical one-dimensional topological defects that may have formed during symmetry-breaking phase transitions in the early universe or in certain string-theoretic scenarios. Their tension is characterized by the dimensionless parameter Gμ, where μ is the energy per unit length and G is Newton's constant. Although no direct evidence for cosmic strings exists, they remain compelling targets for gravitational-wave detectors because oscillating string loops radiate gravitational energy efficiently. A particularly striking signature arises from cusps, regions on a loop that momentarily reach the speed of light and emit sharply beamed bursts of gravitational waves.
The waveform of a cusp burst is highly non-sinusoidal, featuring a characteristic frequency spectrum with amplitude scaling as h(f) ∝ f^{-4/3}. The strain amplitude at a given frequency f observed at a distance r from a loop of length L can be approximated by when the loop size is expressed in seconds (L/c in SI units) and the distance r is in meters. This scaling encapsulates the intense beaming from the cusp: although the total energy emitted is small, the radiation is collimated into a narrow solid angle, producing detectable strains for observers lying within that beam. The factor 2.7 arises from detailed calculations of the Fourier transform of the cusp waveform.
This calculator converts user-specified parameters into an estimated strain amplitude using the formula above. The loop length L is first converted from meters to natural units via division by the speed of light. The distance r is converted from megaparsecs to meters, and the frequency f is taken in hertz. Because the result scales linearly with the dimensionless tension Gμ, even modest uncertainties in μ translate directly into the expected signal strength.
Gravitational-wave observatories operate across a range of frequencies. Ground-based detectors like LIGO and Virgo are most sensitive near 100 Hz, while space-based concepts such as LISA target millihertz frequencies. Pulsar-timing arrays probe nanohertz waves. The frequency dependence in the cusp formula means that shorter loops and higher tensions are better probed at higher frequencies, whereas gigantic loops could produce detectable bursts in the low-frequency bands. By adjusting the frequency and loop length in this calculator, one can explore which detectors are best suited for particular cosmic string scenarios.
The table below offers a sense of how the strain scales with loop size and frequency for a fixed tension Gμ = 10-7 and distance r = 100 Mpc:
| L (m) | f (Hz) | h |
|---|---|---|
| 109 | 100 | ≈ 1.2×10-24 |
| 1010 | 100 | ≈ 5.6×10-24 |
| 1010 | 1000 | ≈ 2.6×10-24 |
| 1011 | 10 | ≈ 1.2×10-23 |
Detection prospects hinge on comparing these strains to the noise curves of current or planned detectors. LIGO's design sensitivity near 100 Hz is roughly h ≈ 10-23, meaning only the largest loops or highest tensions might be marginally detectable. Pulsar-timing arrays, on the other hand, target strains around 10-15 at nanohertz frequencies, potentially probing extremely long loops. The non-detection of cusp bursts thus far places upper limits on Gμ, complementing constraints from stochastic backgrounds and lensing.
Cusps are not the only features of interest on cosmic strings. Kinks—discontinuities in the tangent vector—also emit characteristic bursts with a different spectral slope (h(f) ∝ f^{-5/3}). Interactions between strings can form junctions or lead to loop fragmentation, altering the burst rate and spectrum. A network of cosmic strings in the expanding universe evolves toward a scaling regime where the energy density in strings remains a fixed fraction of the total energy. The distribution of loop sizes and their birth rates determine the expected burst event rate for a given detector. While this calculator focuses on the strain from a single cusp, it can be embedded into broader models to estimate detection rates by integrating over loop populations.
From a theoretical standpoint, cosmic strings arise in grand unified theories when the vacuum manifold has nontrivial first homotopy group, or in string theory when fundamental or D-branes wrap compact cycles. The dimensionless tension Gμ relates to the symmetry-breaking scale η via Gμ ≈ (η² / MPl²). Strings with Gμ ≳ 10-6 would leave clear signatures in the cosmic microwave background, while lighter strings remain viable and could even constitute a component of dark matter. Gravitational-wave bursts provide a window into these high-energy processes long after the early universe has cooled.
Interest in cosmic strings has grown with the advent of high-precision pulsar timing and the first detections of gravitational waves. A network of strings could contribute to the stochastic background observed by pulsar timing arrays, and individual bursts might occasionally stand out above the noise. Numerical simulations of string networks, improved analytic estimates, and Bayesian search techniques all contribute to refining the expected signals and guiding observational campaigns.
By adjusting parameters in this calculator, researchers can rapidly gauge whether a hypothetical cusp burst would fall within the sensitivity of a given detector. The ability to explore extreme values of Gμ and L offers intuition for how cosmic strings might manifest—or remain hidden—in gravitational-wave data. Whether cosmic strings exist or not, the search for their signals continues to sharpen our experimental capabilities and deepen our understanding of the early universe.