  We study one (or two) matrix models modified by terms of the form
$g(\rho(P))^2 + g'(\rho'({\cal{O}}))^2$, where the matrix representation of the
puncture operator $P$ and the one of a scaling operator ${\cal{O}}$ are denoted
by $\rho(P)$ and $\rho'({\cal{O}})$ respectively. We rewrite the modified
models as effective theories of baby universes. We find an upper bound for the
gravitational dimension of ${\cal{O}}$ under which we can fine tune the
coupling constants to obtain new critical behaviors in the continuum limit. The
simultaneous tuning of $g$ and $g'$ is possible if the representations
$\rho(P)$ and $\rho'({\cal{O}})$ are chosen so that the non-diagonal elements
of the mass matrix of the effective theory vanish.
