First-principles microphysical and multi-scale dynamics inputs for firm quantitative constraints on scaling cosmic superstring networks

Cosmic superstrings — F-strings, D-strings and their $(p,q)$ bound states, possibly produced at the end of brane inflation — would form scaling networks whose CMB anisotropies and gravitational-wave backgrounds depend on the fundamental string tension $\mu_F$ and the string coupling $g_s$. Pourtsidou, Avgoustidis, Copeland, Pogosian and Steer (Phys. Rev. D 83, 063525 (2011)) showed that the position of the string-induced CMB B-mode peak and pulsar gravitational-wave bounds constrain different combinations of $(\mu_F,g_s)$, but stressed that firm quantitative constraints are blocked by unresolved inputs: non-perturbative $(p,q)$-string interaction rates, phenomenological compactification-volume factors, the one-scale closure of the network equations, the loop-size distribution and Y-junction gravitational-wave signatures, and the tensor-mode contribution with backreaction. Subsequent progress through 2026 — multitension velocity-dependent one-scale models, GPU simulations of multitension toy networks, junction-aware gravitational-wave spectra, and far stronger LIGO/Virgo and pulsar-timing bounds — has not replaced the phenomenological inputs; every existing quantitative constraint marginalizes over them, and $g_s$ remains unconstrained by observation. The problem of deriving these inputs from first principles and propagating them to firm $(\mu_F,g_s)$ constraints remains open.

Background

Cosmic strings are one-dimensional topological defects that may have formed at symmetry-breaking phase transitions in the early universe; they move relativistically and intercommute (reconnect) when they cross, so a network of long strings and small closed loops settles into a 'scaling' regime in which its energy density stays a fixed fraction of the cosmological total. Such a network would source anisotropies in the cosmic microwave background (CMB) and a stochastic background of gravitational waves (GWs), both actively constrained by observation. Brane-antibrane inflation in type IIB string theory generically ends by producing cosmic superstrings: fundamental F-strings, D1-brane D-strings, and their $(p,q)$ bound states (strings carrying $p$ units of F-string and $q$ units of D-string charge), with a tension spectrum set by the fundamental string tension $\mu_F$ and the string coupling $g_s$. Unlike field-theory strings, superstrings can meet three at a time at Y-junctions (points where three strings of different tensions join, as required by tension balance), and their reconnection probability $P$ can be below unity; both features change the network's evolution and its observational signatures.

Pourtsidou, Avgoustidis, Copeland, Pogosian and Steer (2011) studied multi-tension F-D networks using the multi-tension extension of the velocity-dependent one-scale (VOS) model, the standard analytic framework that describes each string species by a correlation length and an rms velocity evolved with phenomenological momentum-loss and loop-chopping terms (built on Tye, Wasserman and Wyman (2005) and Avgoustidis and Copeland (2010); see also the review by Copeland and Kibble (2010)). They found two scaling regimes: at $g_s\sim 1$ the string stress-energy power spectrum is dominated by populous light F and D strings, while at smaller $g_s$ it is dominated by rare heavy D strings. Because the CMB B-mode spectrum (the curl-like polarization component) sourced by strings peaks at angular scales set by the network correlation length and velocity, this regime change shifts the B-mode peak position, offering in principle the first observational handle on $g_s$; and because CMB and pulsar-timing GW bounds constrain different combinations of string tension and number density, they carve complementary contours in the $(\mu_F,g_s)$ plane.

The source is explicit, however, that these predictions rest on inputs not derived from first principles: (i) the intercommuting and string-production probabilities for collisions of different species, encoded in coefficients $d_{ij}^k$ (for incoming species $i,j$ producing species $k$), which are poorly known in the non-perturbative regime relevant to D-D and heavy-composite collisions; (ii) suppression factors $\mathcal{V}_{ij}(w,g_s)$ representing the possibility that strings moving in compact extra dimensions miss each other, parameterized by a residual compactification parameter $w$; (iii) the one-scale closure that identifies the correlation length along a string with the average inter-string distance, expected to fail when the lightest, dominant species reaches very high number density; (iv) the loop-size distribution (loop length in units of the Hubble radius, usually denoted $\alpha$) together with cusps (loop points momentarily moving at light speed that beam GW bursts) and kinks, all of which enter the GW bounds and are sensitive to Y-junctions; and (v) the tensor-mode (large-scale GW) contribution of strings to the CMB, which requires accounting for backreaction and had not been worked out self-consistently even for ordinary strings. The authors state that follow-up studies on these fronts are needed 'in order to make firmer quantitative predictions' about testing fundamental string theory with CMB and GW observations.

Problem Statement

Replace the phenomenological inputs of multi-tension cosmic superstring network cosmology by first-principles (or controlled-simulation) determinations, and propagate them through network evolution and observational prediction, so that CMB and gravitational-wave data yield firm quantitative constraints on the fundamental string tension $\mu_F$ and the string coupling $g_s$. As stated by the source, this comprises five linked requirements, each feeding the next toward the single objective of firm $(\mu_F,g_s)$ constraints:

(1) Determine the microphysical interaction probabilities — the reconnection and string-production rates encoded in $d_{ij}^k$ for collisions of $(p,q)$ strings of arbitrary species, including D-D and heavy-composite collisions in the non-perturbative regime, with their velocity and angle dependence — from string theory or validated simulations.

(2) Derive the compactification-volume suppression factors $\mathcal{V}_{ij}(w,g_s)$, and hence the residual parameter $w$, from realistic string compactifications rather than phenomenological interpolation.

(3) Extend the network evolution model beyond the one-scale approximation so that it remains valid when the dominant (lightest) species reaches very high number density and its correlation length separates from the average inter-string distance, treating junction dynamics explicitly — via multi-scale analytic modeling or numerical simulation of networks with junctions.

(4) Work out the gravitational-wave signatures self-consistently for multi-tension networks with junctions — the loop-size distribution (parameter $\alpha$), cusp and kink emission, and new waveform contributions from Y-junctions — together with the tensor-mode CMB contribution including backreaction.

(5) Combine (1)-(4) so that CMB constraints (B-mode peak position and amplitude) and gravitational-wave bounds (pulsar timing arrays, ground- and space-based interferometers) on multi-tension networks become firm quantitative constraints on, or measurements of, $(\mu_F,g_s)$, with quantified rather than assumed model dependence.

The five requirements (1)-(5) are jointly posed — the phenomenological inputs (i)-(v) replaced by first-principles or controlled-simulation determinations and propagated through network evolution and observational prediction, so that (5) yields quantified $(\mu_F,g_s)$ constraints.

References

1. A. Pourtsidou, A. Avgoustidis, E. J. Copeland, L. Pogosian, D. A. Steer, 'Scaling configurations of cosmic superstring networks and their cosmological implications', Phys. Rev. D 83, 063525 (2011), DOI 10.1103/PhysRevD.83.063525, arXiv:1012.5014, https://arxiv.org/abs/1012.5014
2. S.-H. H. Tye, I. Wasserman, M. Wyman, 'Scaling of multi-tension cosmic superstring networks', Phys. Rev. D 71, 103508 (2005), DOI 10.1103/PhysRevD.71.103508, arXiv:astro-ph/0503506, https://arxiv.org/abs/astro-ph/0503506
3. A. Avgoustidis, E. J. Copeland, 'The effect of kinematic constraints on multi-tension string network evolution', Phys. Rev. D 81, 063517 (2010), DOI 10.1103/PhysRevD.81.063517, arXiv:0912.4004, https://arxiv.org/abs/0912.4004
4. E. J. Copeland, T. W. B. Kibble, 'Cosmic Strings and Superstrings', Proc. R. Soc. A 466, 623 (2010), DOI 10.1098/rspa.2009.0591, arXiv:0911.1345, https://arxiv.org/abs/0911.1345
5. A. Avgoustidis, E. J. Copeland, A. Moss, L. Pogosian, A. Pourtsidou, D. A. Steer, 'Constraints on the fundamental string coupling from B-mode experiments', Phys. Rev. Lett. 107, 121301 (2011), DOI 10.1103/PhysRevLett.107.121301, arXiv:1105.6198, https://arxiv.org/abs/1105.6198
6. T. Charnock, A. Avgoustidis, E. J. Copeland, A. Moss, 'CMB constraints on cosmic strings and superstrings', Phys. Rev. D 93, 123503 (2016), DOI 10.1103/PhysRevD.93.123503, arXiv:1603.01275, https://arxiv.org/abs/1603.01275
7. P. A. R. Ade et al. (Planck Collaboration), 'Planck 2013 results. XXV. Searches for cosmic strings and other topological defects', Astron. Astrophys. 571, A25 (2014), DOI 10.1051/0004-6361/201321621, arXiv:1303.5085, https://arxiv.org/abs/1303.5085
8. R. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration), 'Constraints on Cosmic Strings Using Data from the Third Advanced LIGO-Virgo Observing Run', Phys. Rev. Lett. 126, 241102 (2021), DOI 10.1103/PhysRevLett.126.241102, arXiv:2101.12248, https://arxiv.org/abs/2101.12248
9. J. J. Blanco-Pillado, K. D. Olum, J. M. Wachter, 'Comparison of cosmic string and superstring models to NANOGrav 12.5-year results', Phys. Rev. D 103, 103512 (2021), DOI 10.1103/PhysRevD.103.103512, arXiv:2102.08194, https://arxiv.org/abs/2102.08194
10. N. Kitajima, K. Nakayama, 'Nanohertz gravitational waves from cosmic strings and dark photon dark matter', Phys. Lett. B 846, 138213 (2023), DOI 10.1016/j.physletb.2023.138213, arXiv:2306.17390, https://arxiv.org/abs/2306.17390
11. S. Datta, R. Samanta, 'Bayesian Forecasts on Cosmic Superstring Searches with LISA' (2026), arXiv:2608.19406, https://arxiv.org/abs/2608.19406
12. I. Yu. Rybak, A. Avgoustidis, C. J. A. P. Martins, 'Dynamics of junctions and the multitension velocity-dependent one-scale model', Phys. Rev. D 99, 063516 (2019), DOI 10.1103/PhysRevD.99.063516, arXiv:1812.04584, https://arxiv.org/abs/1812.04584
13. J. R. C. C. C. Correia, C. J. A. P. Martins, 'Multitension strings in high-resolution U(1)xU(1) simulations', Phys. Rev. D 106, 043521 (2022), DOI 10.1103/PhysRevD.106.043521, arXiv:2208.01525, https://arxiv.org/abs/2208.01525
14. J. R. C. C. C. Correia, 'A New Generation of Cosmic Superstring Simulations', Springer Theses (2023), DOI 10.1007/978-3-031-20229-2, https://doi.org/10.1007/978-3-031-20229-2
15. K. Nakamura, M. Yamada, 'Delayed scaling of multitype cosmic F- and D-strings in velocity-dependent one-scale models', Phys. Rev. D 113, 063537 (2026), DOI 10.1103/PhysRevD.113.063537, arXiv:2512.16821, https://arxiv.org/abs/2512.16821
16. A. Avgoustidis, S. Chadburn, R. Gregory, 'Cosmic superstring trajectories in warped compactifications', Phys. Rev. D 86, 063516 (2012), DOI 10.1103/PhysRevD.86.063516, arXiv:1204.0973, https://arxiv.org/abs/1204.0973
17. T. Elghozi, W. Nelson, M. Sakellariadou, 'Cusps and pseudocusps in strings with Y-junctions', Phys. Rev. D 90, 123517 (2014), DOI 10.1103/PhysRevD.90.123517, arXiv:1403.3225, https://arxiv.org/abs/1403.3225
18. A. Avgoustidis, A. Pourtsidou, M. Sakellariadou, 'Zipping and Unzipping in String Networks: Dynamics of Y-junctions', Phys. Rev. D 91, 025022 (2015), DOI 10.1103/PhysRevD.91.025022, arXiv:1411.7959, https://arxiv.org/abs/1411.7959
19. Y. Matsui, K. Horiguchi, D. Nitta, S. Kuroyanagi, 'Gravitational wave spectrum from kinks on infinite cosmic strings with Y-junctions', J. Cosmol. Astropart. Phys. 11 (2020) 039, DOI 10.1088/1475-7516/2020/11/039, arXiv:2001.01241, https://arxiv.org/abs/2001.01241
20. P. Binétruy, A. Bohé, T. Hertog, D. A. Steer, 'Gravitational Wave Bursts from Cosmic Superstrings with Y-junctions', Phys. Rev. D 80, 123510 (2009), DOI 10.1103/PhysRevD.80.123510, arXiv:0907.4522, https://arxiv.org/abs/0907.4522
21. P. Binétruy, A. Bohé, T. Hertog, D. A. Steer, 'Gravitational wave signatures from kink proliferation on cosmic (super-)strings', Phys. Rev. D 82, 126007 (2010), DOI 10.1103/PhysRevD.82.126007, arXiv:1009.2484, https://arxiv.org/abs/1009.2484
22. S. S. Pufu, V. A. Rodriguez, Y. Wang, 'Scattering from (p, q)-strings in AdS5 x S5', J. High Energy Phys. 03 (2025) 181, DOI 10.1007/JHEP03(2025)181, arXiv:2305.08297, https://arxiv.org/abs/2305.08297
23. L. Sousa, 'Cosmic strings and gravitational waves', Gen. Rel. Grav. 56, 105 (2024), DOI 10.1007/s10714-024-03293-x, https://doi.org/10.1007/s10714-024-03293-x