Let $T$ be a complete theory in a countable language (with only infinite models). Recall the spectrum function: $I(\aleph_\alpha,T)=$ the number of non-isomorphic models of $T$ of cardinality $\aleph_\alpha$. We also recall that $I^*(\aleph_\alpha,T)=|\{M\models T:|M|\leq\aleph_\alpha\}|$ (clearly, we mod-out by isomorphism).
Edit:
Note that $I^*(\aleph_\alpha,T)=|\cup_{\beta \leq \alpha}I(\aleph_\beta,T)|$
Example: Note that if $T$ is dense linear ordering without endpoints or the random graph, then $I^*(\aleph_1,T)=2^{\aleph_1}$
Furthermore, we note that under GCH, we have that if $T$ is uncountably categorical, $I^*(\aleph_{\omega_2},T)=2^{\aleph_1}$
My question is as follows: does there exist some $\kappa$ such that for all $T$ we have $I^*(\xi_T,T)=\kappa$ for some $\xi_T$? What if we assume GCH?
Thanks!