Cosmic String Deficit Angle and Image Separation Calculator

Core cosmic string lensing formulas

A straight cosmic string bends spacetime by removing a wedge rather than by focusing light the way a massive galaxy does. In this idealized picture the line of sight stays locally flat, but the missing wedge creates a conical geometry whose size is set by the dimensionless tension Gμ.

The deficit angle is

Δ = 8π Gμ.

In more explicit mathematical form:

For a source that lies behind the string, the observer sees two undistorted images separated by an angle that depends on how much of the string-to-source baseline sits behind the lens. Using the distances in this 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 cosmic string lensing inputs

String tension Gμ for the cosmic string

The field String Tension Gμ expects a dimensionless number that controls the size of the deficit angle. For this calculator, larger values of Gμ produce proportionally larger image separations. You can type scientific notation such as 1e-6 if that is easier.

String distance D_l (Mpc)

The field String Distance Dl (Mpc) is the observer-to-string distance measured in megaparsecs. Treat it as the lens distance in the simple line-of-sight geometry used here, and keep the same distance convention for D_s.

Source distance D_s (Mpc)

The field Source Distance Ds (Mpc) is the observer-to-source distance in megaparsecs. For the lensing formula on this page to describe a background source, choose values with D_s > D_l so the source sits behind the string rather than in front of it.

The calculator then sets D_ls = D_s − D_l. If you enter D_s ≤ D_l, the computed separation collapses to zero or becomes physically meaningless, which is the correct warning sign for this simplified geometry.

Interpreting the cosmic string lensing results

The output gives you the deficit angle Δ and the angular separation α between the two images created by the string. When reading those numbers, it helps to keep three things in mind:

• Δ (deficit angle): the conical missing wedge around the string. For this model, Δ scales directly with Gμ, so any change in string tension immediately changes the lensing scale.
• α (image separation): the observable split on the sky. It is usually smaller than Δ unless the background source lies far behind the string, because the separation is reduced by the factor D_ls/D_s.
• Radian vs arcsecond: the calculator reports both because sky separations from cosmic strings can be tiny. Astronomers often prefer arcseconds when checking whether an instrument could resolve the double image.

As a rule of thumb:

• α ≃ 1" is resolvable with many ground-based telescopes when the source is compact enough.
• α around 0.1" usually pushes you toward space-based imaging or adaptive optics.
• α at 0.01" or below is very hard to separate from seeing, blending, or source structure.

In this ideal model the two images are mirror copies with no magnification or distortion. The geometry therefore points to a clean doubling rather than the arcs, shear, and magnification patterns that arise in ordinary galaxy or cluster lenses.

Worked cosmic string lensing example

To see the numbers play out, consider a hypothetical straight 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″.

This is a fairly large splitting for a sky image and would be straightforward to resolve in principle, which is why the example is useful for intuition even though realistic cosmic strings are expected to have much smaller tensions.

Comparison with conventional gravitational lenses

Assumptions and limitations of the straight-string model

The cosmic-string lensing calculation on this page is intentionally idealized. The points below explain where the simple geometry is reliable and where it should be treated only as a rough estimate:

• Straight, infinitely thin string: The model treats the lens as a perfectly straight, zero-width string segment. Curvature, kinks, currents, and finite core structure are ignored.
• Single, isolated string: Only one lensing string along the line of sight is included. A real string network could add extra segments or more complicated image patterns.
• Conical metric, no thickness effects: Using Δ = 8πGμ assumes an ideal conical spacetime around the string and leaves out any internal physics of the core.
• Simple distance relation: The calculator uses D_ls = D_s − D_l. That is a useful teaching approximation, but real analyses usually work with angular-diameter distances in an expanding universe.
• Small-angle regime: The split α is assumed to be small. This is appropriate for realistic cosmic-string tensions, but extreme test values should still be interpreted cautiously.
• No magnification or time delay: The page only predicts the angular split between the two images. It does not model flux ratios, lensing shear, or arrival-time differences.
• Not a full survey simulator: Whether a survey can actually identify the double image depends on resolution, noise, source size, foreground clutter, and selection effects that are outside this calculator.
• Use for order-of-magnitude estimates: The results are best treated as quick estimates for cosmic-string searches, classroom examples, or sanity checks against more complete models.

Further context for cosmic string searches

Observational constraints from the cosmic microwave background, pulsar timing, and gravitational-wave searches already narrow the range of viable cosmic-string tensions. By varying Gμ, D_l, and D_s in this calculator, you can see how quickly the angular split grows and where a potential double image might become observable.

That makes the tool handy for planning and teaching: you can pair it with a redshift-to-distance calculator or a broader lensing calculator to build a more realistic sky scenario, then use this page to isolate the distinctive signature of the cosmic string itself.