[Paper Review] On the inverse problem of finding cosmic strings and other topological defects

Matti Lassas, Lauri Oksanen+2 more|arXiv (Cornell University)|May 12, 2015

Numerical methods in inverse problemsMathematics39 references19 citations

TL;DR

This paper develops a microlocal analysis framework to detect spacetime singularities—such as cosmic strings and topological defects—via linearized Cosmic Microwave Background (CMB) measurements. Using the light ray transform on a Friedmann-Lemaître-Robertson-Walker spacetime, it proves that spacelike singularities (moving slower than light) are visible in CMB data, while superluminal singularities are invisible due to gauge invariance, offering a rigorous inverse problem approach to detecting unknown defects without prior geometric assumptions.

ABSTRACT

We consider how microlocal methods developed for tomographic problems can be used to detect singularities of the Lorentzian metric of the Universe using measurements of the Cosmic Microwave Background radiation. The physical model we study is mathematically rigorous but highly idealized.

Motivation & Objective

• To address the inverse problem of detecting spacetime singularities in the Lorentzian metric from CMB measurements.
• To characterize which singularities—such as cosmic strings, monopoles, or cosmic walls—are detectable using tomographic methods.
• To develop a mathematically rigorous, model-agnostic framework that does not assume prior knowledge of defect geometry.
• To identify the role of gauge invariance (diffeomorphisms and conformal changes) in rendering certain singularities invisible to CMB measurements.
• To extend microlocal techniques from medical tomography to cosmological inverse problems, particularly for non-smooth spacetime structures.

Proposed method

• Formulates the inverse problem using a linearized model of CMB measurements, treating the metric's wave front set as the unknown.
• Applies microlocal analysis to the light ray transform on a 1+3 dimensional Friedmann-Lemaître-Robertson-Walker spacetime, restricting to null geodesics.
• Reduces the problem to inverting the light ray transform microlocally, up to potential fields and conformal multiples of the metric.
• Uses the normal operator of the light ray transform and computes its principal symbol as a Fourier multiplier on the cone of spacelike covectors.
• Identifies the kernel of the transform via conformal invariance and gauge invariance, showing that only subluminal singularities contribute to measurable data.
• Employs parametrization techniques and pseudodifferential operator calculus to characterize visibility through the symbol of the normal operator.

Experimental results

Research questions

• RQ1 Which spacetime singularities are detectable in CMB measurements using linearized tomographic methods?
• RQ2 How does the speed of a singularity's propagation (relative to light speed) affect its visibility in CMB data?
• RQ3 What role do diffeomorphism and conformal invariance play in making certain singularities invisible to CMB measurements?
• RQ4 Can microlocal analysis distinguish between visible and invisible singularities without assuming their geometric form?
• RQ5 What is the precise mathematical characterization of the visibility condition for singularities in the context of the light ray transform on cosmological spacetimes?

Key findings

• Singularities moving slower than the speed of light (spacelike covectors) are visible in CMB measurements, as their wave front set appears in the data's wave front set.
• Singularities moving faster than light are invisible because they lie in the kernel of the linearized measurement operator due to gauge invariance.
• The principal symbol of the normal operator of the light ray transform is a non-degenerate Fourier multiplier on the cone of spacelike covectors, enabling microlocal inversion.
• The visibility condition is equivalent to the existence of a light ray intersecting the support of the singularity's cutoff function, ensuring detectability.
• The kernel of the linearized operator includes potential fields and conformal multiples of the metric, corresponding to physical gauge symmetries.
• A parametrix exists for the operator near visible spacelike singularities, confirming that they can be stably reconstructed from CMB data.