Gravitational lensing by cosmic string loops

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Gravitational lensing by cosmic string loops
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Lagrangian of scalar field
Euler-Lagrange equation
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Particular solution
Energy-momentum tensor
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Space-time metric for the case of straight string
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Energy-momentum tensor
Equations of motion
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Lens equations
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Magnification of a point source
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Lens equation for circular loop
Magnification of a point source
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Examples of brightness curves for lensing by
circular loop.
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Lens equations for asymmetric loop
Magnification of a point source
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Caustics (dark), lines of doubling (blue) and
loop (red).
Critical curves (dark) and loop (red).
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Observable data and theoretical curves.
Parameters of the loop
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Position of asymmetric loop at different time.
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Lens equations for binary system
Magnification of a point source
Necessary values of system parameters
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Magnification of stars near ends of caustics
Magnification of stars on cusp of caustics
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Loop distribution
Source distribution
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Lensing probability
For