Cosmic String Formation: Theory & Implications

Updated 2 August 2025

Cosmic string formation is the process where one-dimensional topological defects emerge from early-universe phase transitions and brane interactions.
Field-theoretic mechanisms, notably the Kibble mechanism, describe symmetry breaking that yields vortices with tension proportional to the symmetry breaking scale.
The evolving string networks produce distinctive signatures in CMB anisotropies, gravitational waves, and early structure formation, guiding observational constraints.

Cosmic strings are one-dimensional topological defects formed during early-universe symmetry-breaking phase transitions or, in string theory contexts, as analogs of fundamental extended objects. Their formation mechanisms, dynamic evolution, and cosmological implications are subject to precise theoretical investigation across quantum field theory, string theory, and cosmological modeling frameworks.

1. Field-Theoretic Formation Mechanisms

In gauge field theory, cosmic strings arise when a continuous symmetry is spontaneously broken, giving rise to a vacuum manifold with nontrivial first homotopy group ( $\pi_1(M) \neq 0$ ). A prototypical example involves a complex scalar field $\phi$ coupled to a U(1) gauge field with the Lagrangian

$\mathcal{L} = |D_\mu \phi|^2 - \frac{1}{4} F_{\mu\nu}F^{\mu\nu} - V(\phi)$

where $D_\mu = \partial_\mu + i e A_\mu$ and $V(\phi) = \lambda(|\phi|^2 - \eta^2)^2$ (0911.1345).

At high temperatures, $\langle\phi\rangle = 0$ , but after cooling below a critical temperature, the field acquires a vacuum expectation value $|\phi| = \eta$ , and the vacuum manifold becomes $S^1$ . Spatial regions separated by more than the correlation length choose random phases, enabling nontrivial windings around closed loops. The Kibble mechanism rigorously prescribes that a nonzero winding around a spatial loop necessitates a vortex — a cosmic string — through the loop (0911.1345).

Typically, the Nielsen–Olesen vortex ansatz for a straight string along the $z$ -axis is

$\phi(r,\theta) = \eta f(r) e^{i n \theta}, \quad A_\theta(r) = - \frac{n}{e} h(r)$

with winding number $n$ . The field equations for $f(r), h(r)$ with $f(0)=0$ , $f(\infty)=1$ , $h(0)=0$ , $h(\infty)=1$ specify the core structure and energy profile. The tension (energy per unit length) is typically set by the symmetry breaking scale: $\mu \sim \eta^2$ (0911.1345).

2. The Kibble Mechanism and Scaling Network Formation

The Kibble mechanism ensures that, due to causality, phase choices of the Higgs field in disconnected regions are uncorrelated, forcing the system to develop defects — cosmic strings — at the interfaces. As a consequence, at the time of formation $t_s \sim (G\mu)^{-1} t_p$ (where $t_p$ is the Planck time), the string network consists of $\mathcal{O}(10)$ long strings per Hubble volume plus a stochastic distribution of loops produced by intercommutation and self-intersection (Vilenkin et al., 2018).

The network rapidly approaches a scaling solution, characterized by invariant statistical properties under rescaling with the cosmic horizon. The long-string energy density scales as $\rho_s(t) \sim 10 \mu / t^2$ (Vilenkin et al., 2018), and loop production maintains a scale-invariant distribution, as captured by the Polchinski–Rocha model: $\frac{dn}{d\alpha} = \frac{\mathcal{C}(\alpha)}{\alpha d_H^3}$ where $\alpha$ is the loop size parameter (relative to the horizon), $\mathcal{C}(\alpha)$ is a power law $\sim \alpha^{-p}$ with $p=1.41$ –1.60 depending on the cosmological era (1006.0931).

3. String Theory Scenarios: Superstring and Composite String Formation

In string theory and M-theory, post-inflationary dynamics such as brane–antibrane annihilation offer alternative formation channels (Copeland et al., 2011). F-strings, D-strings, and bound $(p,q)$ -strings can emerge when tachyon condensation breaks world-volume symmetry. The tension spectrum is

$\mu_{(p,q)} = \frac{1}{2\pi \ell_s^2} \sqrt{p^2 + q^2/g_s^2}$

where $p$ , $q$ count F and D-string charge, $\ell_s$ is the string length, and $g_s$ the string coupling (Copeland et al., 2011).

Nontrivial junctions ("Y-junctions") commonly form when strings of different type collide, as in $(p, q_1)+(p',q_2) \rightarrow (p+p',q_1+q_2)$ bound states (0804.0200). The reconnection probability can be much less than unity (unlike the field-theory expectation of $P\sim1$ ); for F-strings, $P \sim g_s^2 \ll 1$ , and for D-strings or composite strings the suppression can be exponential (Yamada et al., 2022). This modulates network properties, allowing denser superstring networks.

4. Junction Dynamics and Composite Structures

Y-junctions arise when two strings of different type interact, frequently forming a bridge — a composite string segment — with a pair of Y-junctions at its ends (0804.0200). The formation, late-time evolution, and velocity of such composite strings are governed by both field-theoretic simulations (e.g., coupled $U(1)\times U(1)$ Higgs models) and the effective Nambu–Goto action with Lagrange multiplier-imposed junction conditions.

Explicitly, for a “cosine connectivity” encounter, the bridge velocity $u$ satisfies the quartic

$0 = u^4 S^2 \sin^2\alpha + u^2 [ R^2 (1-v^2) + S^2 (v^2 \cos^2\alpha - \sin^2\alpha) ] - S^2 v^2 \cos^2\alpha$

where $R = \mu_3/(\mu_1+\mu_2)$ , $S = (\mu_1-\mu_2)/(\mu_1+\mu_2)$ , $v$ is the collision velocity, and $\alpha$ the incidence angle.

The orientation and growth rate of the bridge (connecting the two original strings) are given by

$\tan\theta = \frac{u}{v} \tan\alpha, \quad s_3 = \frac{u\cos\alpha - R\gamma\cos\theta}{\gamma\cos\theta - Ru\cos\alpha}$

with $\gamma = 1/\sqrt{1-v^2}$ (0804.0200). Simulations show that after a brief radiative burst upon formation, the late-time regime closely matches these Nambu–Goto dynamics, with negligible energy loss to radiation and persistent Y-junction stability.

5. Alternative and Nonconventional Formation Channels

Mechanisms beyond the global SSB scenario have been developed. Notably, evaporating primordial black holes (PBHs) can locally heat the plasma via Hawking radiation to above a symmetry restoration threshold; as the plasma then expands and cools, local SSB occurs, generating string loops in the vicinity of each PBH (Srivastava, 6 May 2024). If the density of PBHs is adequate and their local hot spots overlap, string intercommutation leads to percolation and the formation of an infinite network — even if the overall cosmological temperature never allows a full SSB transition. The requisite percolation condition is $d_\text{PBH} < R_\text{stretch}$ , where the left side is the typical PBH separation and $R_\text{stretch}$ is the maximum distance to which loops are dragged by plasma outflows.

A further novel scenario is percolation-driven formation from small initial loops with decreasing tension $\mu(t)$ in the context of string cosmological kination. If $2 H + \dot\mu/\mu < 0$ , loops grow faster than the scale factor and eventually percolate to form an interconnected network (Conlon et al., 18 Jun 2024). For compactification scenarios with volume moduli rolling toward large values (e.g., LVS), this can yield current-day fundamental string networks with phenomenologically interesting tension $G\mu \sim 10^{-10}$ .

6. Early Structure Formation and Astrophysical Impact

Cosmic strings and their associated loops seed nonlinear, non-Gaussian perturbations that can drastically influence the early structure formation history. In both analytic (Shlaer et al., 2012, Jiao et al., 2023) and N-body simulation approaches (Jiao et al., 9 Feb 2024), string loops rapidly accrete matter, forming highly elongated, filamentary halos whose mass function scales as a power law in $(1+z)$ , in contrast to the exponentially suppressed high redshift bins of Gaussian $\Lambda$ CDM. For string tensions $G\mu \gtrsim 10^{-8}$ , the abundance of string-induced halos can dominate at $z \gtrsim 12$ , matching stellar mass densities inferred by JWST for early galaxies.

The presence of cosmic strings can also drive unique signatures in the CMB (e.g., Kaiser–Stebbins discontinuities, B-mode polarization) (0911.1345, Copeland et al., 2011), stochastic gravitational wave backgrounds (1006.0931, Yamada et al., 2022), and, through Y-junctions and composite structures, can yield novel signatures such as modified gravitational lensing (Y-shaped images) and nontrivial superstring network dynamics (0804.0200, Copeland et al., 2011).

7. Extensions, Stability, and Observational Constraints

Observed cosmic string tension is constrained at $G\mu\lesssim 10^{-7}$ by CMB observation (0911.1345, Copeland et al., 2011), with gravitational wave and pulsar timing analyses pushing bounds to $G\mu\lesssim 10^{-9}$ for small-loop dominated networks. Stability of the network is preserved since energy loss due to radiative emission during Y-junction formation and late-time dynamics is negligible; networks tend toward a scaling regime in which their energy density remains a fixed fraction of the total.

Hybrid scenarios such as cosmic strings with black holes as beads (Ashoorioon et al., 2014) or in multi-step symmetry-breaking results (Chitose et al., 18 Jun 2025) introduce richer network topology, stages of metastable and stable strings, and the potential for novel monopole or flux-carrying endpoint structures — with important implications for the detailed spectrum and decay products of the network.

The precise microphysical parameters (tension, reconnection probability, symmetry-breaking pattern) and cosmological context (reheating temperature, PBH abundance, modulus dynamics) determine the resultant network evolution, its observable signatures, and its impact on early structure formation.

Table 1: Canonical Formation Mechanisms and Key Properties

Scenario	Defect Type	Key Property / Distinction
U(1) gauge symmetry breaking	Topological string	Tension $\mu\sim\eta^2$
Brane (F/D-string) annihilation post-inflation	(p,q)-string	Tension $\mu_{(p,q)}$ ; Y-junctions possible
PBH evaporation-induced local SSB	String loops	Network percolates via hot-spot overlap
Kination-driven loop growth	Expanding loops	$2H+\dot\mu/\mu<0$ : growing, percolating

Cosmic string formation is a phenomenon with rigorously predicted structural, dynamical, and cosmological consequences, governed both by microphysical (symmetry breaking, field dynamics, brane interactions) and astrophysical (phase transitions, PBH population, expansion history) parameters. The latest theoretical, numerical, and observational research delineates a clear picture in which the mechanisms of string formation and network evolution are critical for understanding the signatures potentially observable through gravitational waves, CMB observations, and early cosmic structure.