The following may not be an answer to your question, but I think it is related. I have taken it from the introduction of a joint work I am doing with James Cummings and Sy Friedman (which has now appeared):
An important development in large cardinal theory is the construction of inner models $M$ all of whose sets are definable from ordinals and which serve as good approximations to the entire universe $V$. The former means that $M$ is contained in $HOD$, the universe of hereditarily ordinal definable sets, and the latter can be interpreted in a number of ways. One such interpretation is that the cardinal structure of $M$ is ``close'' to that of $V$ in the sense that $\alpha^+$ of $M$ equals $\alpha^+$ of $V$ for many cardinals $\alpha.$ This is for example the case if $V$ does not contain $0^{\sharp}$ and $M$ equals $L$, or if $V$ does not contain an inner model with a Woodin cardinal and $M$ is the core model $K$ for a Woodin cardinal.
The following theorem shows that we can't hope to approximate the cardinals of $V$ by those of (inner models of) $HOD$ in general:
Theorem Suppose $GCH$ holds and $\kappa$ is a $\kappa^{+4}-$supercompact cardinal. Then there is a generic extension $V^*$ of $V$ in which $\kappa$ remains inaccessible and for all infinite cardinals $\alpha <\kappa,$ $(\alpha^{+})^{HOD}<\alpha^{+}.$ In particular $W=V_{\kappa}^{V^{*}}$ is a model of $ZFC$ in which for all infinite cardinals $\alpha, (\alpha^{+})^{HOD}<\alpha^{+}.$
