The loop-formation size distribution and modelling systematics of the cosmic-superstring stochastic gravitational-wave background

Cosmic superstring networks are among the few string-theory scenarios with observable consequences, and their stochastic gravitational-wave background (SGWB) can be fit to pulsar-timing data. A 2025 study by Avgoustidis, Copeland, Moss and Raidal showed that a multi-tension network model fits the NANOGrav 15-year signal at likelihoods comparable to supermassive black-hole binaries, but identified the size of loops at formation (the parameter $\alpha_i$ for each string species $i$) as the single most important unknown shaping both the amplitude and the low-frequency spectral shape, alongside linked systematics: approximate self-/cross-interaction coefficients in the multi-tension velocity-dependent one-scale model, gravitational backreaction and a possibly time-varying emitted power $\Gamma$, the model-dependent mapping from compactification geometry to the effective volume parameter $w$, and the treatment of string junctions. The problem is to determine the loop-formation size distribution (including its species dependence) from the dynamics of multi-tension $(p,q)$ string networks, and to quantify how the remaining modelling systematics propagate into the predicted SGWB, so that the background becomes a precision probe of the underlying string-theory parameters.

Background

Cosmic strings are one-dimensional topological defects that can form during symmetry-breaking phase transitions in the early universe and stretch to cosmological scales; they radiate predominantly gravitational waves from closed oscillating loops that chop off the network. Cosmic superstrings arise in certain string-theory scenarios where fundamental strings (F-strings) and one-dimensional D-branes (D-strings) are stretched to cosmic scales during brane inflation. Because string coupling allows bound states, the network contains multiple species, the $(p,q)$ strings carrying $p$ units of F-string and $q$ units of D-string charge, with tension $\mu_{(p,q)}=(\mu_F/g_s)\sqrt{p^2 g_s^2+q^2}$, where $\mu_F$ is the fundamental tension and $g_s$ the string coupling. Unlike ordinary cosmic strings, which reconnect with probability one when they intersect, superstrings intercommute with reduced, channel-dependent probabilities set by the microscopic string theory, and colliding strings can also form Y-shaped three-string junctions (so-called zipper segments) rather than simply passing through or exchanging partners.

Network evolution is modelled with the velocity-dependent one-scale (VOS) approach, which tracks a correlation length $L_i$ and root-mean-square velocity $v_i$ per species; its multi-tension extension adds self-interaction coefficients $\tilde{c}_i$ and cross-interaction coefficients $\tilde{d}_{ij}^k$ describing energy transfer between species. Loops are born with length $\ell_i(t_b)=\alpha_i t_b$ at birth time $t_b$, and the parameter $\alpha_i$ is empirically unknown: simulations of ordinary Nambu-Goto strings (the standard zero-thickness string model whose dynamics extremize the worldsheet area) by Blanco-Pillado, Olum and Shlaer found a double-peaked loop production spectrum with roughly 90% of the energy going into loops near the gravitational-backreaction scale $\alpha\sim 2\Gamma G\mu$ (where $\Gamma\sim 50$ is the total gravitational-wave power emitted by a loop in units of $G\mu^2$) and about 10% into large loops with $\alpha\sim 0.1$, but it is not known whether these results transfer to multi-tension superstring networks, where the reduced intercommutation and junction dynamics can change the small-scale structure. Individual loops emit gravitational waves at harmonics of their fundamental frequency, with spectra that differ depending on whether the emission is dominated by cusps (points on the loop instantaneously moving at the speed of light) or kinks (discontinuities in the loop tangent), and this cusp/kink content is itself affected by gravitational backreaction, which recent simulations show can smooth loops and suppress cusps and can make the emitted power $\Gamma$ time-dependent.

Pulsar timing arrays (PTAs) — collaborations that monitor millisecond pulsars to detect nanohertz gravitational waves — such as NANOGrav (the North American Nanohertz Observatory for Gravitational Waves) have found evidence for a stochastic nanohertz gravitational-wave background, and both the NANOGrav 15-year new-physics analysis and the work of Figueroa, Pieroni, Ricciardone and Simakachorn found that cosmic-string models fit the signal well once the loop size is left free. Against this backdrop, Avgoustidis, Copeland, Moss and Raidal (2025) evolved a multi-tension VOS superstring network with up to seven species, computed the full nanohertz-to-kilohertz SGWB, and fit it to the NANOGrav 15-year data, obtaining well-constrained fundamental tensions ($\log_{10}(G\mu_1)\approx -11.4$ for large loops, $\approx -9.7$ for small loops) but flat posteriors on the string coupling $g_s$ and on a parameter $w$ describing the effective volume of the compact extra dimensions, which controls the intercommutation probabilities of heavy strings. The qualitative difference between the large-loop and small-loop spectra — a smooth low-frequency tail versus a sharp multi-peaked structure with the heavier species dominating — makes the loop-size question decisive for interpreting current and future data: the smoking-gun multi-tension features lie at the lowest frequencies, probing of which requires roughly thirty years of PTA observation time, while the high-frequency tails are accessible to LISA (the Laser Interferometer Space Antenna, a planned space-based gravitational-wave detector) and the Einstein Telescope (a planned third-generation ground-based gravitational-wave detector).

Problem Statement

Determine, from the dynamics of multi-tension cosmic-superstring networks, the loop-formation size distribution — the full loop production function per species, characterized by the typical loop-birth sizes $\alpha_i$ for each $(p,q)$ string species, their species dependence, and the spread of the production spectrum around the typical values — and quantify how the remaining modelling systematics of the multi-tension VOS description propagate into the predicted stochastic gravitational-wave background. The linked systematics to be quantified are: (i) the accuracy of the power-law parameterizations used for the self- and cross-interaction coefficients $\tilde{c}_i$ and $\tilde{d}_{ij}^k$ for realistic compactification choices; (ii) the effect of gravitational backreaction on loop evolution, including a possibly time-varying total emitted power $\Gamma$, on the loop distribution and the harmonic (cusp/kink) content of the spectrum; (iii) the mapping from microscopic compactification geometry and warping to the effective volume parameter $w$ and the channel-dependent intercommutation probabilities; and (iv) the consequences of the assumed junction-formation and zipper-segment dynamics (including the energy-partition parameter $B$ and the segment lengths $\ell_{ij}$).

These components serve one overarching objective, stated by the source: to replace the current practice of scanning over discrete loop-size scenarios (small loops $\alpha_i\sim 2\Gamma G\mu_i$ versus large loops $\alpha_{L,i}\equiv \ell_i/L_i\sim 0.37$; the two $\alpha$ parameters are normalized differently — $\alpha_i$ to cosmic time $t$ and $\alpha_{L,i}$ to the correlation length $L_i$ — so that $\alpha_{L,i}\sim 0.37$ reproduces the simulation value $\alpha\sim 0.1$ for ordinary strings) with a derived prediction, so that the multi-tension SGWB spectrum, with quantified uncertainties, becomes a precision probe of the underlying string-theory parameters — the fundamental tension $\mu_F$, the string coupling $g_s$, and the effective size of the compact extra dimensions through $w$ — in current and planned gravitational-wave observations. The problem is theoretical and numerical: the loop production function per species is to be established, and the propagated uncertainty of the spectrum is to be quantified with respect to each of the systematics (i)–(iv), rather than bracketed by ad hoc scenario choices or handled only qualitatively.

References

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