The study of isomorphism spectra of structures - the collection of Turing degrees realized by isomorphic copies of a structure - has been a central topic in computable structure theory in the last decades. Recently researchers started to look at degree spectra under relations other than isomorphism, such as spectra of theories and $\Sigma_n$ spectra. We investigate another type of spectrum, the elementary bi-embeddability spectrum of a structure.

We obtain examples of families of degrees which form elementary bi-embeddability spectra of structures but are not spectra of theories and also give examples of theories whose spectrum can not be realized as the elementary bi-embeddability spectrum of any structure.

On our quest to distinguish elementary bi-embeddability spectra from isomorphism spectra we consider possible invariants and obtain examples of families of degrees which can not be realized as isomorphism spectra of structures with special model theoretic properties such as prime models and homogeneous structures.