Cosmic Strings

A crack in the frozen vacuum

Cool water below 0 °C and it freezes into ice — but it rarely freezes into one flawless crystal. Different regions start crystallising independently, each picking its own orientation, and where mismatched domains meet they leave permanent flaws: grain boundaries (2D), dislocation lines (1D), point vacancies (0D). The flaw is trapped because the lattice on either side has already committed, and no smooth rearrangement can heal it.

The early universe did something analogous. As it expanded and cooled through about 10⁻³⁶ seconds, the vacuum is thought to have undergone phase transitions in which a symmetric field configuration broke to a less symmetric one — the way a Higgs-type field rolls from zero to a non-zero value. Causally disconnected patches each picked an independent value of the field. Wherever the field could not be smoothly combed flat across a boundary, it got trapped in the old high-energy phase along a line. That trapped filament of unbroken vacuum, threading the universe, is a cosmic string. It is the cosmological cousin of a vortex line in superfluid helium-4 or a flux tube in a type-II superconductor.

The Kibble mechanism and when strings form

Tom Kibble formalised this in 1976. Whether a phase transition makes topological defects, and of what dimension, is decided entirely by the topology of the vacuum manifold — the set of degenerate lowest-energy field configurations after breaking. If that manifold is not simply connected (it has non-contractible loops, π₁ ≠ trivial), the theory produces strings. A few canonical cases:

Vacuum manifold topology	Non-trivial homotopy	Defect produced	Dimension
Disconnected (π₀ ≠ 0)	π₀	Domain walls	2D sheet
Has non-contractible loops (π₁ ≠ 0)	π₁	Cosmic strings	1D line
Has non-contractible 2-spheres (π₂ ≠ 0)	π₂	Magnetic monopoles	0D point
π₃ ≠ 0	π₃	Textures	delocalised

The simplest string-producing model is a complex scalar field with a "Mexican-hat" potential, where the vacuum manifold is a circle. As the field settles, its phase winds by an integer multiple of 2π around any loop encircling the string; that winding number cannot change continuously, so the core where the field is forced back to zero is topologically locked in. The Kibble estimate is that the network forms with roughly one string segment per causal horizon at the transition, giving an initial correlation length comparable to the horizon size.

The physics: a conical deficit, not a force

The defining strangeness of a cosmic string is its gravity. A static, straight string has equal energy density and tension along its length, so in the energy-momentum tensor the time-time and the along-the-string space-space components are equal and opposite. Plug that into Einstein's equations and the gravitational potential cancels: a test particle initially at rest beside the string feels no Newtonian acceleration toward it.

Yet the string is not gravitationally invisible. It removes a wedge of space. The metric around a straight string is locally flat but globally that of a cone — a flat sheet with a pie-slice cut out and the edges glued. The missing angle is the deficit angle:

Δ = 8π Gμ / c²

where μ is the mass per unit length and the everything-else is the dimensionless tension. For a GUT-scale string with Gμ/c² ≈ 10⁻⁶, the deficit angle is

Δ = 8π × 10⁻⁶ rad ≈ 2.5 × 10⁻⁵ rad ≈ 5.2 arcseconds

Two light rays passing the string on opposite sides travel through ordinary flat space, but the cone tilts the two paths toward each other by Δ. To an observer, a single galaxy directly behind the string splits into two undistorted, equal-brightness images separated by a few arcseconds — gravitational lensing with no lens and no magnification gradient. This is qualitatively unlike the arcs and rings produced by a massive galaxy or cluster (see gravitational lensing).

The key numbers

Cosmic strings live at scales that bracket the absurd. The two parameters that matter are the symmetry-breaking energy scale η and the dimensionless tension Gμ/c² ≈ (η / M_Planck)², where the Planck energy is 1.22 × 10¹⁹ GeV.

Quantity	GUT-scale string (η ~ 10¹⁶ GeV)	Comment
Dimensionless tension Gμ/c²	~ 10⁻⁶	(η/M_Pl)²; now ruled out
Mass per unit length μ	~ 10²² g/cm ≈ 10²¹ kg/m	≈ mass of Earth per ~4 km
Core width	~ 10⁻³⁰ cm	set by η⁻¹, ≈ 10⁻¹⁷ × a proton radius
Deficit angle Δ	~ 5 arcsec	8πGμ/c²
Formation epoch	t ~ 10⁻³⁶ s, T ~ 10²⁹ K	around the GUT transition
Allowed today (PTA limit)	Gμ/c² ≲ 10⁻¹¹	pulsar-timing arrays, 2023

The numbers are staggering: a GUT-scale string thinner than a proton by seventeen orders of magnitude still carries about the mass of the Earth in every few kilometres of its length. A loop a kilometre across would outweigh a mountain. Crucially, these are the original GUT-scale numbers; the actual upper limits today are far smaller, which is why most of the interest has migrated from "did strings build the galaxies?" to "can we hear their gravitational waves?"

From a tangle to a scaling network

A single string is a curiosity; the cosmological object is the network — and the plural in "cosmic strings" matters. Just after formation the universe is filled with a random tangle of long, wiggly strings plus a soup of closed loops, with a correlation length set by the horizon. Left to redshift passively, this network's energy would dilute more slowly than matter or radiation and could come to dominate the universe, over-closing it — a disaster.

It doesn't, because of two dynamical facts. First, when two strings cross they intercommute: they swap partners and reconnect, with a probability close to 1 for ordinary field-theory strings. Self-intersections of a single long string chop off closed loops. Second, those loops oscillate relativistically and radiate their energy away — mostly as gravitational waves. The combined effect drives the network to a remarkable attractor first identified in 1980s simulations (Albrecht & Turok; Bennett & Bouchet) called the scaling solution: at every epoch there are just a handful of long strings crossing each Hubble volume, and the fraction of the universe's energy in strings stays constant and small. The network looks statistically identical at every time when measured in units of the horizon. This self-similar behaviour is what makes a string network compatible with the universe we see.

Worked example: the gravitational-wave amplitude

Decaying string loops are the loudest thing a network does. Estimate the gravitational-wave background they leave today. A loop of length ℓ oscillates with period ℓ/2c and radiates power

P_GW = Γ G μ² c    with the dimensionless constant Γ ≈ 50

so a loop's lifetime is its energy μℓc² divided by that power:

τ = μ ℓ c² / P_GW = ℓ / (Γ G μ / c²)
  = ℓ / (Γ · Gμ/c²)

Take a loop born at matter-radiation equality with ℓ ~ 10⁻³ of the horizon and a tension Gμ/c² = 10⁻¹¹. With Γ = 50, the loop survives Γ⁻¹ (Gμ/c²)⁻¹ ≈ 2 × 10⁹ oscillation lengths' worth of Hubble time — long enough to keep radiating across cosmic history while it slowly shrinks. Summed over the whole population and all epochs, the predicted background is roughly flat in the dimensionless spectral density Ω_GW(f) over many decades of frequency, with amplitude

Ω_GW h² ∼ 10⁻⁷ to 10⁻⁹   for Gμ/c² in the 10⁻¹⁰ to 10⁻¹¹ range

That broadband flatness — extending from nanohertz (pulsar timing) up through the LISA millihertz band and toward the LIGO–Virgo–KAGRA hundred-hertz band — is the smoking-gun spectral fingerprint that distinguishes a string background from the steeper spectrum expected of merging supermassive black-hole binaries.

How we hunt for them

No cosmic string has been confirmed. Four complementary searches set the running limits:

• CMB anisotropies. A string moving across the sky imprints a temperature step on the cosmic microwave background — the Kaiser–Stebbins effect, ΔT/T ~ 8πGμ/c² × (v/c). Strings produce incoherent, non-Gaussian, vector- and tensor-mode fluctuations and cannot make the sharp acoustic peaks. Planck (2013–2015) used the angular power spectrum and non-Gaussianity to cap the string contribution to a few tenths of a percent of the temperature power, i.e. Gμ/c² ≲ 10⁻⁷. See the cosmic microwave background.
• Gravitational lensing. Search the sky for pairs of identical galaxy images with no distortion gradient. The 2003 candidate CSL-1 generated excitement before Hubble imaging in 2006 resolved it into two separate elliptical galaxies — a cautionary tale, not a detection.
• Pulsar-timing arrays. The strongest current probe. A stochastic background perturbs the arrival times of millisecond-pulsar pulses with the characteristic Hellings–Downs angular correlation. The 2023 NANOGrav 15-year dataset (plus EPTA, PPTA and CPTA) reported evidence for a nanohertz background; a string network with Gμ/c² ~ 10⁻¹¹–10⁻¹⁰ fits comparably to the astrophysical black-hole-binary interpretation. Earlier non-detections already pushed Gμ/c² below ~10⁻¹¹ for the standard loop model.
• Burst gravitational waves. Cusps (points that momentarily reach the speed of light) and kinks on loops emit sharp, beamed GW bursts. LIGO–Virgo–KAGRA search for these individual transients; no detection yet, with limits complementary to the stochastic searches.

Discovery, the inflation rivalry, and key people

The idea grew from condensed-matter analogy. Tom Kibble (1976) worked out which grand-unified phase transitions produce which defects and estimated their cosmological abundance. Yakov Zeldovich (1980) and Alexander Vilenkin (1981) recognised that GUT-scale strings, with Gμ/c² ~ 10⁻⁶, had just the right density contrast to seed galaxies — a tantalising coincidence that made strings a serious competitor to inflation as the origin of cosmic structure throughout the 1980s.

That rivalry was settled by the CMB. Inflation predicts coherent, adiabatic, near-scale-invariant fluctuations that produce the sharp acoustic peaks; strings, being constantly stirred and incoherent, smear those peaks out. COBE (1992), then decisively WMAP (2003) and Planck (2013–2018), measured peaks so sharp that strings cannot be the dominant seed. The field reframed: strings became a possible sub-dominant relic and, increasingly, a gravitational-wave target. The 2023 PTA announcements put them back in the spotlight. A parallel thread came from string theory itself — Edmund Witten noted in 1985 that fundamental superstrings would over-produce defects, but later work (Polchinski and others, c. 2004) showed that "cosmic superstrings" (fundamental F-strings and D-strings stretched to cosmic size) can survive with intercommutation probabilities below 1, giving the network a distinctive observational handle.

Variants and relatives

• Cosmic superstrings. Fundamental or Dirichlet strings from string theory, stretched to cosmological scale by inflation. Their intercommutation probability can be as low as 10⁻³, which makes the network denser and the GW background louder for a given tension — a target signature for LISA and PTAs.
• Superconducting strings (Witten, 1985). Strings carrying a conserved current. They couple to electromagnetism, can radiate light and particles as well as gravitational waves, and could power radio bursts or seed primordial magnetic fields.
• Local vs global strings. Local (gauge) strings have all their energy confined to the core; global strings have a long-range Goldstone-field halo and radiate predominantly in light scalar particles (e.g. axions) rather than gravitational waves — relevant to axion dark matter cosmology.
• Domain walls and monopoles. The 2D and 0D siblings from the same Kibble bestiary. Walls are cosmologically dangerous (they over-close the universe unless they decay); monopoles are the subject of the monopole problem that inflation was partly invented to solve.
• Vortons. Hypothetical stable loops of superconducting string held open by the angular momentum of their trapped current — a potential dark-matter candidate if they survive.

Common misconceptions and subtleties

• "A cosmic string pulls you in." A straight static string exerts zero Newtonian force. Its effect is purely the conical deficit angle. (Two parallel moving strings, or a wiggly string, do gain an effective attraction — but the textbook straight, static string does not.)
• "Cosmic strings are the same as the strings in string theory." Different objects. Ordinary cosmic strings are macroscopic field-theory solitons. Cosmic superstrings are a separate, more exotic possibility in which fundamental string-theory strings happen to be stretched to cosmic size — they share phenomenology but not origin.
• "Strings seeded the galaxies." They were a leading candidate in the 1980s but were ruled out as the dominant seed by the sharp CMB acoustic peaks. Inflation won that contest; strings survive only as a constrained sub-dominant relic.
• "They have ends." Topological cosmic strings are either infinite (stretching across the universe) or closed loops. A string cannot simply terminate in empty space, because the winding of the field around it has nowhere to unwind. (Cosmic superstrings can end on D-branes, an exception that distinguishes them.)
• "The deficit angle means space is curved near the string." Space around a straight string is locally flat — the curvature is concentrated entirely on the (infinitely thin) string itself, like the apex of a cone. The global geometry is conical; the local geometry is Euclidean everywhere off the string.