  We present a general relation between celestial correlation functions in
$d$-dimensions and Witten diagrams in $\left(d+1\right)$-dimensional Euclidean
anti-de Sitter (EAdS) space, to all orders in perturbation theory. Contact
diagram processes are proportional to contact Witten diagrams, and particle
exchanges can be recast as a continuum of particle exchanges in EAdS where the
exchanged particles carrying unitary Principal Series representations of
$SO\left(d+1,1\right)$. One can then try to import familiar EAdS techniques to
study the properties of celestial correlators. In this work we use this
relation to infer the analytic structure of the spectral density in the
conformal partial wave expansion of celestial correlators which, at least
perturbatively, should be a meromorphic function of the spectral parameter. We
also discuss non-perturbative constraints from unitarity in Euclidean Conformal
Field Theory, which requires positivity of the spectral density. This extends
similar relations recently uncovered between boundary correlation functions in
de Sitter space and Witten diagrams in EAdS, suggesting that EAdS could play a
central role in efforts towards holography for all lambdas.
