Self-consistent dynamics and radiation of axion-emitting cosmic superstring networks

Benabou, Bonnefoy, Buschmann, Kumar and Safdi (Phys. Rev. D 110, 035021 (2024)) showed that if a string-theory axion acquires a cosmic string network in the early Universe — for example from D-brane annihilation at the end of D-brane inflation — the string tension is parametrically larger than that of field-theory axion strings, so the network overproduces axion dark matter except in strongly warped or multi-axion ('axiverse') scenarios. However, every network quantity entering that conclusion — the loop-size distribution, the axion-versus-gravitational-wave power split, the emission spectral index, and their dependence on the intercommutation probability $P<1$ — was conjectured by porting field-theory global-string results and Nambu-Goto superstring simulations, the two existing but disjoint numerical literatures. The paper states that dedicated simulations of cosmic superstrings that self-consistently account for axion production have never been performed. The open problem is to determine, by self-consistent simulation of a hybrid network of thin, high-tension D-/F-string cores with $P<1$ coupled to the four-dimensional axion field they source, the scaling regime, loop spectrum, and instantaneous axion and gravitational-wave emission spectra — including the effective parameters $r_a$, $r_G$, $q$, and $\alpha$ — on which the cosmological conclusions rest.

Background

The axion is a light pseudoscalar particle proposed to solve the strong CP problem of QCD (the unexplained smallness of charge-parity violation in the strong interactions, bounded by searches for a neutron electric dipole moment); in field theory it is the pseudo-Nambu-Goldstone boson of a spontaneously broken global Peccei-Quinn (PQ) symmetry, characterized by a decay constant $f_a$. If PQ symmetry breaks after inflation, a network of axion strings forms. Because the broken symmetry is global, the axion phase winds around each string core and the associated field gradients extend out to the Hubble radius, so the strings continuously radiate their energy into axion waves; such strings are called global (as opposed to local, gauge) strings. Field-theory global axion-string networks have been simulated on the lattice for decades (e.g. Buschmann et al., arXiv:2108.05368; Kaltschmidt, Redondo, Saikawa and Vaquero, arXiv:2502.02398), fixing, for example, the number of long strings per Hubble patch in the scaling regime — the attractor state in which the network's properties per Hubble volume become approximately time-independent — and the shape of the instantaneous axion emission spectrum, the latter remaining actively debated.

In string theory, axions arise generically not from PQ fields but as zero modes of higher-dimensional gauge forms (e.g. the ten-dimensional NSNS and RR two-forms) integrated over cycles of the compact internal space. Benabou, Bonnefoy, Buschmann, Kumar and Safdi (Phys. Rev. D 110, 035021 (2024)) asked what happens if such a string-theory axion has a cosmic string in its cosmological history. Such strings are not PQ-field-theory objects: they are fundamental strings (F-strings) or D-branes — membrane-like extended objects on which open strings can end — wrapped on internal cycles (D-strings) stretched across the four large dimensions — collectively, cosmic superstrings — produced for instance by brane-antibrane annihilation at the end of D-brane inflation. Their tension is set by the string and compactification scales and is parametrically larger than $f_a^2$ (except for cycles at the bottom of strongly warped regions). From the axion's viewpoint these strings look like global strings — the axion field winds around the core — while the core itself is thin and essentially fundamental, unresolvable in the low-energy effective field theory.

This hybrid character makes their network dynamics unlike either simulated limit. First, when two superstrings cross they reconnect (intercommutate) with probability $P$ that can be well below 1 — D-string collisions reconnect with $P\sim0.1$–$1$, and colliding strings can also miss each other by separating in an extra compact dimension — whereas field-theory strings have $P\approx1$. Nambu-Goto simulations, which approximate strings as infinitely thin objects that move so as to minimize their world-sheet area and include no radiation channel (Avgoustidis and Shellard, arXiv:astro-ph/0512582; Urrestilla and Vilenkin, arXiv:0712.1146), found that the long-string density is approximately independent of $P$ for $P\gtrsim0.1$, a result the source paper imports. Second, the network loses energy in two channels: to gravitational waves, dominated by loop production, with a per-loop power $dE_{GW}/dt=r_G\,G\mu^2$ and $r_G\approx50$ (here $G$ is Newton's constant and $\mu$ the string tension, i.e. the energy per unit length), and to axion radiation, for which the paper conjectures a per-loop power $dE_a/dt=r_a f_a^2$ with $r_a\sim\mathcal{O}(10)$ ported from global-string simulations. The instantaneous emission spectra are assumed to be power laws $\propto\omega^{-q}$ with $1<q<2$ ($q=4/3$ for cusp-dominated and $5/3$ for kink-dominated loop emission, where a cusp is a point on a loop that momentarily moves at the speed of light and a kink is a sharp corner created when strings intercommutate), while modern global-string simulations tend to find $q\approx1$. These inputs — the loop-production function normalization $\alpha$ within the velocity-one-scale (VOS) model, a semi-analytic framework evolving the average string separation and velocity, together with $r_a$, $r_G$, $q$, and their $P$-dependence — propagate into the paper's conclusions: unwarped string-theory axion strings overproduce QCD-axion dark matter for every allowed axion mass, while in a multi-axion 'axiverse' — the scenario in which the compactification yields many axions with a wide range of masses — a heavier axion mass eigenstate (making the string a higher-tension 'axiverse string' in the sense of March-Russell and Tillim, arXiv:2109.14637) can collapse the network early, restoring a viable abundance and producing potentially observable gravitational waves. The low-frequency end of the axion emission spectrum, at energies $\omega$ near or below the axion mass $m_a$, fixes what fraction of emitted axions is nonrelativistic at network collapse and is handled only with a cutoff.

Problem Statement

Construct and evolve a self-consistent model of an axion-carrying cosmic-superstring network — thin, high-tension D-/F-string cores with intercommutation probability $P<1$ (including the extra-dimensional miss channel), coupled to the four-dimensional axion field whose phase winds around them — through the radiation-era scaling regime, and determine from it the quantities that Benabou et al. (Phys. Rev. D 110, 035021 (2024)) had to conjecture, namely: (i) the scaling density of long strings and the loop-length distribution, i.e. the loop-production function and its normalization $\alpha$, and how they depend on $P$; (ii) the instantaneous differential axion and gravitational-wave emission spectra, in particular the spectral index $q$ and the behavior at frequencies near and below the axion mass $m_a$, which controls the nonrelativistic fraction of emitted axions at network collapse; and (iii) the effective emission parameters, the per-loop axion power $r_a$ (conventionally normalized through $dE_a/dt=r_a f_a^2$) and gravitational-wave power $r_G$, and their ratio, as functions of $P$. These quantities are the coupled outputs of one network, and the objective they jointly serve is to replace the semi-analytic inputs of the source paper — ported from $P=1$ field-theory global strings and from Nambu-Goto superstring simulations without radiation — with measured or controlled values, so that its cosmological conclusions (overproduction of QCD-axion dark matter by unwarped string-theory axion strings; the viability and gravitational-wave signal of the axiverse heavy-eigenstate collapse loophole) are placed on quantitative footing.

References

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