Cosmic String Gravitational Lensing Calculator

Overview: What this calculator does

This tool estimates the gravitational lensing signature of an ideal, straight cosmic string. Given the string tension Gμ and the comoving or physical distances (in Mpc) from the observer to the string (D_l) and to the background source (D_s), it computes:

• The spacetime deficit angle Δ created by the string.
• The angular separation α between the two images of a background source, in radians and arcseconds.

The calculator implements a simple geometric model of cosmic string lensing, suitable for quick estimates, teaching, and order-of-magnitude checks against observational limits.

Core formulas used

A long, straight cosmic string does not focus light like a massive galaxy. Instead, it creates a conical spacetime: a wedge of angle is removed and the edges are identified. This produces a deficit angle Δ that depends only on the dimensionless string tension Gμ, where G is Newton’s constant and μ is the mass per unit length of the string.

The deficit angle is

Δ = 8π Gμ.

In more explicit mathematical form:

For a source located behind the string, the observer sees two images separated by an angle α that depends on the geometry. Using the distances in the calculator:

• D_l: distance from observer to the string (Mpc).
• D_s: distance from observer to the background source (Mpc).
• D_ls: distance from the string to the source, approximated by D_ls = D_s − D_l.

Under the small-angle and thin-string approximations, the image separation is

α = Δ · (D_ls / D_s).

The calculator evaluates:

1. Compute the deficit angle in radians:
   Δ = 8π Gμ.
2. Compute the string–source distance:
   D_ls = D_s − D_l.
3. Compute the image separation in radians:
   α = Δ (D_ls / D_s).
4. Convert α from radians to arcseconds:
   α_arcsec = α × 206265.

How to enter inputs

String tension Gμ (dimensionless)

The field String Tension Gμ expects a dimensionless number. In many particle-physics and cosmology models, interesting values lie in the range Gμ ≈ 10⁻¹⁰ – 10⁻⁶. Current observational constraints typically limit Gμ to ≦ 10⁻⁷–10⁻⁶, depending on the analysis.

You can enter scientific notation using standard calculator syntax (for example, 1e-6 for 10⁻⁶ if your interface supports it).

String distance D_l (Mpc)

The field String Distance Dl (Mpc) is the distance from the observer to the cosmic string, measured in megaparsecs (1 Mpc ≈ 3.26 million light-years). You may treat this as a comoving or physical distance, as long as you use the same convention for D_s.

Source distance D_s (Mpc)

The field Source Distance Ds (Mpc) is the distance from the observer to the background source, again in megaparsecs. For the geometric formula above to make sense, you should choose values with D_s > D_l, so that the source lies behind the string.

The calculator then sets D_ls = D_s − D_l. If you enter D_s ≤ D_l, the image separation will be zero or unphysical, reflecting that the source is not actually behind the string along the line of sight.

Interpreting the results

The output provides the deficit angle Δ and the angular separation α between the two images. Key points for interpretation:

• Δ (deficit angle): a global property of spacetime around the string. Larger Gμ leads directly to a larger Δ.
• α (image separation): the observable angular distance between the two images on the sky. It is smaller than Δ if the string is not very close to the source (because α is scaled by D_ls/D_s).
• Radian vs arcsecond: astronomers usually quote very small angles in arcseconds. For example, 1 arcsecond (1″) ≈ 4.848×10⁻⁶ radians.

As a rule of thumb:

• α ≡ 1″ (one arcsecond) is resolvable with many ground-based telescopes and good seeing.
• α ≈ 0.1″ requires space telescopes or adaptive optics.
• α ≤ 0.01″ is extremely challenging to detect directly in imaging surveys.

In this simple model, the two images are unmagnified and undistorted mirror copies of each other. Their combined brightness is the same as that of the unlensed source, unlike conventional gravitational lenses (galaxies, clusters), which typically magnify and stretch images into arcs.

Worked example

Suppose you consider a hypothetical cosmic string with Gμ = 1 × 10⁻⁶, located at D_l = 1000 Mpc, with a background galaxy at D_s = 2000 Mpc.

1. Compute the deficit angle:
   Δ = 8π Gμ ≈ 8π × 10⁻⁶ ≈ 2.51 × 10⁻⁵ radians.
2. Compute D_ls:
   D_ls = 2000 − 1000 = 1000 Mpc.
3. Image separation in radians:
   α = Δ (D_ls / D_s) = 2.51 × 10⁻⁵ × (1000 / 2000) ≈ 1.26 × 10⁻⁵ rad.
4. Convert to arcseconds:
   α_arcsec ≈ 1.26 × 10⁻⁵ × 206265 ≈ 2.6″.

A separation of roughly 2.6 arcseconds would be well within the resolving power of many modern telescopes, making this an observationally promising case, provided such a high-tension string actually existed and aligned with a bright source.

Comparison with conventional gravitational lenses

Assumptions and limitations

The calculation here is intentionally simple and idealized. Keep the following assumptions and caveats in mind when interpreting results:

• Straight, infinitely thin string: The model assumes a perfectly straight, infinitely thin cosmic string. Real strings may have small-scale structure, curvature, or form loops, which can modify the lensing pattern.
• Single, isolated string: Only one string segment along the line of sight is considered. Networks of strings or multiple segments are not modeled.
• Conical metric, no thickness effects: The use of Δ = 8πGμ assumes an ideal conical spacetime and neglects any internal structure or finite width of the string core.
• Simple distance relation: The geometry uses D_ls = D_s − D_l. In a full cosmological treatment, angular-diameter distances in an expanding universe would be used more carefully. Here, you supply distances consistently, and the code applies this linear approximation.
• Small-angle regime: The formula for α is valid for small angles, which is well satisfied for realistic values of Gμ but should be remembered when testing extreme, nonphysical values.
• No magnification or time delay: The model predicts no magnification, no shear, and no differential time delay between the two images. It only returns the angular separation, not detailed light curves or flux ratios.
• Not a full survey simulator: Detectability depends on telescope resolution, signal-to-noise, source structure, and confusion from nearby objects. A small predicted separation does not guarantee that a particular instrument can resolve the images.
• Use for order-of-magnitude estimates: The results are best interpreted as order-of-magnitude or pedagogical estimates. Precision cosmology or detailed comparisons with real survey data require more complete modeling.

Further context and use

Constraints on Gμ from cosmic microwave background anisotropies, pulsar timing arrays, and gravitational-wave detectors already limit the allowed parameter space for cosmic strings. By exploring different values of Gμ, D_l, and D_s with this calculator, you can see which combinations would yield image separations large enough to be observable.

For teaching and research planning, you can combine this tool with general gravitational lensing or cosmology calculators (for example, to convert redshifts to distances) to build more realistic scenarios. The simple output here—deficit angle and image separation—is a useful starting point for understanding how a cosmic string would manifest itself in imaging surveys.