Geometry of Cosmic String Lensing

Cosmic strings are hypothetical one-dimensional topological defects that may have formed during symmetry-breaking phase transitions in the early universe. Unlike ordinary gravitational lenses such as galaxies or black holes, a long straight cosmic string does not focus light by its mass; instead, it alters the global topology of spacetime. The spacetime around an ideal string is locally flat but globally conical—the string removes a wedge of angle from space and stitches the edges together. Light rays passing on opposite sides of the string travel in straight lines, yet because the space has a missing angle, their apparent directions differ for a distant observer. This effect is known as gravitational lensing by a cosmic string. In its simplest manifestation, an unresolved background source appears duplicated, with the two images separated by an angle related directly to the string’s tension. The absence of magnification or time delay between images makes detection challenging, but the distinctive pattern of two identical images offers a promising signature if the separation lies within the resolving power of telescopes.

The deficit angle generated by a string is determined by its dimensionless tension parameter Gμ, where G is Newton’s constant and μ is the mass per unit length. In natural units with c = 1, the spacetime metric around an infinitely thin string yields a conical geometry characterized by the deficit angle Δ = 8πGμ. If Gμ is on the order of 10⁻⁶, as predicted by some grand unified theories, the deficit angle is several arcseconds—sufficiently large to separate images in optical surveys. For strings with lower tension, the separation shrinks linearly, making detection progressively more difficult. The calculator uses the relation Δ = 8πGμ to compute the fundamental geometrical effect. Users supply the value of Gμ, often constrained by cosmic microwave background measurements and gravitational wave searches, along with the distance to the string D_l and the distance to the background source D_s. Assuming the observer, string, and source are nearly aligned, the angular separation between the two images is approximately

$\alpha =\Delta \frac{D{\mathrm{ls}}_{}}{D{s}_{}}$

where D_ls = D_s − D_l is the distance from the string to the source. The result α is the separation in radians between the two images. The script converts this value to arcseconds for astronomical relevance. Because the lensing is purely geometrical, the two images are undistorted mirror copies; there is no magnification, so the combined flux equals that of the unlensed source. This contrasts sharply with conventional lensing by massive objects, where images are magnified, stretched into arcs, and separated by deflection angles depending on the mass distribution. The peculiar signature of cosmic string lensing—duplicated, un-magnified images—serves as a beacon for cosmologists searching the sky for relics of high-energy physics.

To gain a quantitative feeling for the lensing effect, consider the following numerical example. Suppose Gμ = 10⁻⁶, the string lies at D_l = 1000 Mpc, and the background galaxy is at D_s = 1500 Mpc. The deficit angle is then Δ ≈ 2.5×10⁻⁵ radians. The image separation follows as α ≈ Δ × (500/1500) ≈ 8.3×10⁻⁶ radians, or about 1.7 arcseconds. Many wide-field surveys achieve sub-arcsecond resolution, meaning such a string could produce easily resolvable double images. If the tension were an order of magnitude smaller, the separation would drop to roughly 0.17 arcseconds, demanding higher resolution instruments such as the Hubble Space Telescope or upcoming extremely large telescopes. The table below summarizes these examples and illustrates how the image separation scales with tension and geometry.

Gμ	Dl (Mpc)	Ds (Mpc)	Separation (arcsec)
1×10-6	1000	1500	1.7
1×10-7	1000	1500	0.17
5×10-7	500	1000	0.82

The observations of such image pairs would have profound implications. Detecting a cosmic string would directly probe energy scales far beyond those accessible in terrestrial laboratories, possibly revealing the nature of symmetry breaking in the early universe. Strings may also form networks whose intersections produce loops that radiate gravitational waves, contributing to stochastic backgrounds sought by pulsar timing arrays. If cosmic string lensing is observed, follow-up searches for associated gravitational wave signals or high-energy emissions from cusp events could provide a multifaceted picture of these exotic objects. Conversely, the absence of lensing signatures in high-resolution surveys constrains Gμ, complementing limits from the cosmic microwave background and gravitational wave experiments. This synergy exemplifies how multiple astrophysical probes combine to test particle physics theories beyond the Standard Model.

The conical spacetime around a string implies additional subtle effects. For instance, a moving string can induce line discontinuities in the cosmic microwave background temperature due to the Kaiser-Stebbins effect. Such signatures have motivated dedicated searches in CMB maps. Similarly, arrays of cosmic strings could imprint distinctive patterns on large-scale structure by seeding wakes in the cosmic matter distribution. While the calculator focuses on the simple lensing geometry of a single straight string, the underlying principles extend to these more complex phenomena. In scenarios with junctions or loops, the deficit angles and resulting lensing patterns become richer, involving cusp caustics and multiple image pairs. Understanding the basic relation between tension and image separation forms the foundation for analyzing these advanced configurations.

From a practical standpoint, the calculator converts the input distances from megaparsecs to meters using the factor 3.08567758 × 10²² m/Mpc. It then computes D_ls, evaluates the deficit angle Δ = 8πGμ, and finally determines the separation α. The result is displayed in arcseconds by multiplying the radian value by 206265. The interface is intentionally minimalistic to avoid overwhelming new users, yet the underlying computation remains accurate for a wide range of cosmological distances and tension values. Users seeking to explore survey detectability can vary Gμ and the geometry to see how the separation compares with the angular resolution of instruments.

While cosmic strings remain hypothetical, the intellectual exercise of calculating their lensing signatures nurtures intuition about how topology and geometry interplay in general relativity. By experimenting with different parameters, users can appreciate how even a negligible mass per unit length can leave an observable imprint through global spacetime structure. This calculator therefore serves not only as a numerical tool but also as an educational platform, inviting curious minds to contemplate the deep connections between high-energy physics, cosmology, and gravitational lensing.