Cosmic microwave background (CMB) is the thermal radiation remnant from the Big Bang and is a primary source of information regarding the early universe. An outstanding question is what information can be inferred from it. In this talk, we discuss some results on recovering gravitational perturbations of scalar type. First, we consider the (integrated) Sachs-Wolfe effect which leads to an X-ray tomography problem in the cosmological setting. Second, to include photon absorption and scattering, we consider the Boltzmann equation model and study an inverse source problem in which the source term is connected to the metric perturbation. Finally, we address nonlinear effects in Nordstrom’s scalar gravitation theory. Our analysis relies on the microlocal properties of the cosmological X-ray transform (or light ray transform), and its interplay with hyperbolic PDEs.
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