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New Massless Spectra from Cosmic String Cusps

Amelia Drew, Ivan Rybak

TL;DR

This work expands the canonical picture of cosmic-string cusps by classifying non-generic cusp shapes and deriving corresponding massless radiation spectra. Using Taylor expansions around cusp points and travelling-wave configurations, the authors show that generic cusps yield the familiar $ ilde{κ}(ω) ∝ ω^{-4/3}$ tail, while non-generic cusps admit a family of spectra with exponents determined by the lowest nonzero derivatives on the left- and right-moving modes, namely $ ilde{κ}(ω) ∝ ω^{-(β_n+β_m)}$ with $β_n=n/(2n-1)$ and $n,m≥2$. They extend the Damour–Vilenkin framework to handle cusps with derivative discontinuities and zeroes of lower-order derivatives, uncovering new power laws and demonstrating how realistic corrections (backreaction, finite width) impose high-frequency cutoffs via Airy-function dependent expressions. The results provide richer, testable templates for gravitational-wave and axion signals from cosmic strings, informing both theoretical modeling and observational searches while highlighting the role of UV physics in shaping cusp phenomenology.

Abstract

In the standard picture of cosmic strings, cusps are generic features of Nambu-Goto loops where the string momentarily reaches the speed of light. They have a characteristic sharp profile, following $y \sim x^{2/3}$ in the $(x,y)$ plane, and produce strong gravitational-wave (GW) bursts with frequency-domain strain $\mathop{\tilde{\!κ}}(ω) \propto ω^{-4/3}$, making them key targets for current and future GW searches. However, under certain conditions, this generic picture can differ. We identify cusp solutions with different, including smooth, shapes, and compute their massless GW and axion spectra. We derive a general expression for all possible Nambu-Goto cusp spectra with the asymptotic form $\mathop{\tilde{\!κ}}(ω) \propto ω^{-n/(2n-1)} ω^{-m/(2m-1)}$ where $n, m\geq 2$. We investigate the effect of realistic corrections to the Nambu-Goto description, such as those from backreaction and finite string width, which introduce a high frequency cutoff.

New Massless Spectra from Cosmic String Cusps

TL;DR

This work expands the canonical picture of cosmic-string cusps by classifying non-generic cusp shapes and deriving corresponding massless radiation spectra. Using Taylor expansions around cusp points and travelling-wave configurations, the authors show that generic cusps yield the familiar tail, while non-generic cusps admit a family of spectra with exponents determined by the lowest nonzero derivatives on the left- and right-moving modes, namely with and . They extend the Damour–Vilenkin framework to handle cusps with derivative discontinuities and zeroes of lower-order derivatives, uncovering new power laws and demonstrating how realistic corrections (backreaction, finite width) impose high-frequency cutoffs via Airy-function dependent expressions. The results provide richer, testable templates for gravitational-wave and axion signals from cosmic strings, informing both theoretical modeling and observational searches while highlighting the role of UV physics in shaping cusp phenomenology.

Abstract

In the standard picture of cosmic strings, cusps are generic features of Nambu-Goto loops where the string momentarily reaches the speed of light. They have a characteristic sharp profile, following in the plane, and produce strong gravitational-wave (GW) bursts with frequency-domain strain , making them key targets for current and future GW searches. However, under certain conditions, this generic picture can differ. We identify cusp solutions with different, including smooth, shapes, and compute their massless GW and axion spectra. We derive a general expression for all possible Nambu-Goto cusp spectra with the asymptotic form where . We investigate the effect of realistic corrections to the Nambu-Goto description, such as those from backreaction and finite string width, which introduce a high frequency cutoff.

Paper Structure

This paper contains 17 sections, 50 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Review of Nambu-Goto Dynamics
  3. Shapes of Cusps
  4. Generic Cusps
  5. Non-generic Cusps
  6. Discontinuous $\mathbf{X}_{{\rm c} \, \pm}^{(n)}$
  7. $\mathbf{X}_{{\rm c} \pm}^{(p)}=0$ for all $1<p<n$
  8. Explicit Examples with Travelling Waves
  9. Massless Radiation Spectra
  10. Gravitational and Axion Radiation from Strings
  11. Damour and Vilenkin Spectrum
  12. Spectra of Cusps with Discontinuous $\mathbf{X}_{{\rm c} \pm }^{(2)}$
  13. New Cusp Spectra
  14. More Realistic Cusps
  15. Discussion
  16. ...and 2 more sections

Figures (5)

  • Figure 1: The $y\sim x^{2/3}$ shape of the generic cusp in physical space, where the dot represents a point moving at the speed of light, and corresponding Kibble-Turok sphere. The direction of radiation exhibiting a $\tilde{\kappa}(\omega) \propto \omega^{-4/3}$ spectrum is schematically indicated by the dashed arrow and corresponding emission cone.
  • Figure 2: The shape of a smooth cusp in physical space, where the discontinuous function $\textbf{X}_{\rm c}^{(2)}$ smooths out the shape and the dot represents a point moving at the speed of light. The right panel represents the Kibble-Turok sphere for this cusp realisation. The direction of radiation exhibiting a $\tilde{\kappa}(\omega) \propto \omega^{-4/3}$ spectrum is schematically indicated by the dashed arrow and corresponding emission cone.
  • Figure 3: Dynamics of the string that at the time $\tau=0$ in the middle of illustrated segment with $\sigma=0$ develops a cusp with $X^{(2)} = 0$, which is described by equation \ref{['CrazyCusp']}. The Kibble-Turok sphere is the same as in Figure \ref{['Fig:Cusp2']}. The direction of radiation exhibiting a $\omega^{-(\beta_n+\beta_m)}$ spectrum, where $\beta = \frac{n}{2n - 1}$ and $n \in \mathbb{Z},\ n > 2$, is schematically indicated by the dashed arrow and the associated emission cone.
  • Figure 4: The left panel shows the shape of a travelling wave cusp, as described by Eq. \ref{['TravelWave']}. The right panel depicts the corresponding Kibble–Turok sphere for this cusp configuration. No radiation is emitted for this particular setup.
  • Figure 5: Radiation spectra computed using \ref{['ThickCusp']} with cusp deviation defined by \ref{['CuspDevi']}. Lines for $\Delta_{\pm} = 10^{-1}$ to $10^{-4}$ are compared to the dashed $|I^\mu_\pm|\propto \omega^{-2/3}$ power law. Smaller $\Delta_{\pm}$ values shift the cutoff to higher frequencies.