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Recent Lyman-$\alpha$ forest tomography measurements of the intergalacticmedium (IGM) have revealed a wealth of cosmic structures at high redshift($z\sim 2.5$). In this work, we present the Tomographic AbsorptionReconstruction and Density Inference Scheme (TARDIS), a new chrono-cosmographicanalysis tool for understanding the formation and evolution of these observedstructures. We use maximum likelihood techniques with a fast non-lineargravitational model to reconstruct the initial density field of the observedregions. We find that TARDIS allows accurate reconstruction of smaller scalestructures than standard Wiener filtering techniques. Applying this techniqueto mock Lyman-$\alpha$ forest data sets that simulate ongoing and futuresurveys such as CLAMATO, Subaru-PFS or the ELTs, we are able to infer theunderlying matter density field at observed redshift and classify the cosmicweb structures. We find good agreement with the underlying truth both in thecharacteristic eigenvalues and eigenvectors of the pseudo-deformation tensor,with the eigenvalues inferred from 30m-class telescopes correlated at $r=0.95$relative to the truth. As an output of this method, we are able to furtherevolve the inferred structures to late time ($z=0$), and also track thetrajectories of coeval $z=2.5$ galaxies to their $z=0$ cosmic web environments.

TARDIS Paper I: A Constrained Reconstruction Approach to Modeling the z~2.5 Cosmic Web Probed by Lyman-alpha Forest Tomography