New Massless Spectra from Cosmic String Cusps

The gist: This work identifies cusp solutions with different, including smooth, shapes, and computes their massless GW and axion spectra from Nambu-Goto string dynamics.

Review of Nambu-Goto Dynamics

The classical evolution of a one-dimensional cosmic string in the limit of infinitely thin string width can be modelled using the Nambu-Goto effective action (Page 3). The action is given by S = −µ √−γ d2ζ. (1) The equation of motion in a flat Minkowski background, using the conformal gauge X˙ µX′µ = 0, reduces to wave equations X¨ µ − X′′µ = 0 (Page 3). A general solution for the string motion is given by X0 = τ, X(τ, σ) = ½ (X+(σ+) + X−(σ−)) (4) where σ± = τ ± σ and X±(σ±) are 3-dimensional vectors that denote components moving in opposite directions along the string (Page 3). The evolution of the string can therefore be described by trajectories of X(1)± on a 2-dimensional unit sphere, known as the Kibble-Turok sphere (Page 3).

Shapes of Cusps

The shape of a cusp in real space can be calculated by Taylor expanding the real space position vector X around the cusp point Xc (Page 5). For generic cusps, this expansion yields for each mode X±(σ±) = Xc± + ∑X(n)c± σn/n! (8) where the parameterisation is chosen such that the cusp occurs at σc± = 0 (Page 5). This leads to the string core tracing the curve y ∼ x2 /3 in the (x, y) plane near the cusp point, as shown in Figure 1 (Page 5).

Non-generic cusps arise under different conditions:

1. Discontinuous X(n)c±: To produce a smooth cusp, the function X(n)c± must change sign at the cusp point such that lim σ→σ+ c X(n)± (σ±) = − lim σ→σ- c X(n)± (σ±) ≠ 0 (Page 6). This condition produces a smooth cusp contour as shown in Figure 2 (Page 6).

2. X(p)c± = 0 for all 1 < p < n: In this case, one can generally choose a coordinate system in which the n-th and (n + 1)-th derivatives lie in the (x, y) plane, generically producing a cusp shape of y ∼ xn/(n+1) (Page 7).

Massless Radiation Spectra

The calculation of massless spectra is reduced to evaluating an integral derived from the Fourier transform of the source terms (Page 9). For generic cusps, substituting the Taylor expansion into equation (28) leads to an analytic expression for the asymptotic behaviour of this integral when ω → ∞ using the saddle point approximation (Page 10). This results in a spectrum κ˜(ω) ≈ ω −4/3 power law (Page 11).

The general frequency dependence for cusps with lowest non-zero derivative terms n and m is given by κ˜(ω), ˜bµν(ω) ≈ ω −(βn+βm), n, m > 1, n, m ∈ Z (Page 12). This reveals a far richer structure of possible signatures than previously recognised (Page 14).

More Realistic Cusps

For cosmic strings arising from spontaneously broken symmetries, the Nambu-Goto equations of motion do not hold exactly (Page 15). Higher-order corrections, such as those from finite string thickness or backreaction effects, can modify the string dynamics and may prevent the formation of cusps (Page 15). These corrections are captured by introducing approximate solutions X¯(1)± to incorporate small deviations from Nambu-Goto induced by the effects discussed above (Page 15).

When cusp conditions are no longer satisfied exactly, lower-order terms are retained and affect the spectrum (Page 41). The resulting asymptotic waveform is given by Iµ± = 2πiΦ X¯(2)µc± sgn(γ±) 3/2ω2 /3 γ± 2/3 iβ± ω −1/3 (3 γ±) 2/3 Ai(ζ±) + Ai′(ζ±)!, where setting ∆± = β± = 0 recovers the result for generic cusps (Page 14). The presence of a current on cosmic strings can suppress cusp formation, including in the idealised limit of zero string thickness (Page 17).

Discussion

A general consensus has emerged around the features of cusps that are important for determining their massless emission, namely that they are points on a loop of string that travel at the speed of light, around which the string has a sharp shape y ∼ x2 /3 and which generate massless radiation with a κ˜(ω) ∼ ω −4/3 power law spectrum (Page 15). This work suggests that even in the Nambu-Goto limit, cusp configurations do not always radiate with a κ˜(ω) ∼ ω −4/3 power law spectrum (Page 15). The spectra derived in Section 4.4 fill the range between ω −4/3 and ω −1, with discrete power laws of the form κ˜(ω) ∼ ω −(n2n−1 + m2m−1), where n, m ≥ 2 are positive integers given by the lowest non-zero derivative terms in the Taylor expansion around the cusp point for each mode (Page 14). The connection between cusp geometry and its massless spectrum need not follow this conventional intuition; not only can cusps have other shapes, but the massless spectrum is not directly tied to the geometric shape in real space (Page 15).

The analysis suggests that deviations from Nambu-Goto behaviour in field-theoretic strings need not be confined to sharply localised features (Page 15). For the smooth cusp configurations investigated here, backreaction from massive vector radiation will dominate over massless backreaction, and this will reduce the overall strength of the signal (Page 15). This work suggests a potential avenue for future simulations of cosmic string cusps based on the collision of travelling waves, such as those presented in Section 3 (Page 15).

The final conclusion is that cusp configurations can exhibit shapes ranging from sharp profiles to completely smooth ones, and the massless spectrum is not directly tied to the geometric shape in real space (Page 15). The predicted spectrum for n = 1 corresponds to the predicted frequency dependence κ˜(ω) ∼ ω −2 of a signal from a kink-kink collision (Page 15).

The paper concludes that deviations from Nambu-Goto behaviour will lead to some high frequency cutoff in the signal associated with the energy or length scale of the relevant effect, which would modify the predicted spectrum both for individual bursts and stochastic background signals from a string network (Page 15). The results are generalized to ‘realistic’ field theory strings in Section 5 (Page 15).

The work is supported by several acknowledgements including support from the European Union under the Horizon Europe research and innovation programme, Marie Sklodowska-Curie Project 101151409 GWStrings, and a Junior Research Fellowship at Homerton College, University of Cambridge (Page 16). The paper is based upon work from COST Action COSMIC WISPers CA21106 (Page 16).

The paper is available on arXiv:2512.12258v1 [hep-ph] 13 Dec 2025 (Page 4). The references include works such as T. W. B. Kibble, T. Damour and A. Vilenkin, and various papers on gravitational waves from cosmic strings (Page 18).

The paper is structured to review Nambu-Goto dynamics in Section 2, outline the method to calculate cusp shapes in Section 3, and review the calculation of massless spectra in Section 4 (Page 2). The results are generalized to ‘realistic’ field theory strings in Section 5 (Page 6). The discussion and conclusion are presented in Section 6 (Page 2).

The paper provides explicit examples with travelling wave configurations on long strings to give concrete examples of cusp configurations (Page 7). These examples can be used to verify the massless spectra for the new cusps presented, including the dependence on string parameters of the position of the high frequency cutoff that comes from corrections to pure Nambu-Goto (Page 15). The paper is available on arXiv:2512.12258v1 [hep-ph] 13 Dec 2025 (Page 4).

The paper is available on arXiv:2512.12258v1 [hep-ph] 13 Dec 2025 (Page 4). The references include works such as T. W. B. Kibble, T. Damour and A.

1. Improve gravitational wave search algorithms by incorporating a wider range of cusp spectral shapes. The system can now distinguish between generic cusps yielding a sharp profile, following y ∼ x 2/3 and non-generic shapes like those arising from discontinuous X(n)c ± or the smooth cusp contour described by equation (13).

2. Enhance axion search sensitivity by modeling the full spectrum of massless radiation. The improved system can now predict spectra of New Cusp Spectra, specifically capturing the frequency dependence where κ˜(ω) ∼ ω −(n 2n−1 + m 2m−1), which includes novel decay rates like κ˜(ω) ∼ ω −3/5 when X(2)c ± = 0.

3. Develop real-time waveform analysis capabilities to differentiate between Nambu-Goto predictions and realistic field theory effects. The system can now incorporate corrections from finite string width or backreaction, allowing it to predict a high frequency cutoff in the spectrum based on the scale of these physical deviations, as shown in Figure 5.

4. Implement a predictive framework for signal suppression mechanisms by simulating departures from the ideal Nambu-Goto limit. The AI can now model how nonzero string width or massless backreaction lead to slight deviations from that predicted in the strict Nambu-Goto limit, resulting in a modification of the spectrum based on parameters like ∆±, β± and γ±.