Cosmic-string cusp beams: closed forms for the beam solid angle and the backreaction cutoff (technical note)

We study the gravitational-wave beam of a single cusp on a Nambu-Goto cosmic-string loop in the local Airy description, and report four results. (i) The detector-weighted beam solid angle obeys an exact scaling, Omega_eff = C (s_a s_b)^(1/3) / (n^(2/3) sin psi), where s_a and s_b are the slopes of the two tangent curves at their crossing on the Kibble-Turok sphere and psi is their crossing angle. The sin psi dependence is implicit in the azimuthal integral of Wachter, Olum and Blanco-Pillado; what is given here is its closed form and the detection-weighted version. (ii) A parameter-free on-axis n^(-2/3) correction to the leading cusp amplitude, validated against exact Fourier integrals on 297 cusps of 75 loops with rounded kinks and replicated on 491 cusps of 80 random-harmonic loops. (iii) The cusp beam for a small speed deficit Delta, as produced by gravitational backreaction: the 1/sin psi dependence survives while the slope scaling breaks, validated on 14 cusp geometries. (iv) The resulting change to the burst-rate cutoff, an order-one refinement of the recently derived on-axis cutoff. All tests use constructed loops; no real loop catalogue was available. The conversion from Delta to G mu is loosely calibrated, so the burst-rate statements are illustrative. This is an unreviewed technical note and the code is research code not checked by specialists. Computations were assisted by Claude (Anthropic). Code: https://github.com/siddharaj93/cuspbeam