  We consider the double scaling limit for a model of $n$ non-intersecting
squared Bessel processes in the confluent case: all paths start at time $t=0$
at the same positive value $x=a$, remain positive, and are conditioned to end
at time $t=1$ at $x=0$. After appropriate rescaling, the paths fill a region in
the $tx$--plane as $n\to \infty$ that intersects the hard edge at $x=0$ at a
critical time $t=t^{*}$. In a previous paper (arXiv:0712.1333), the scaling
limits for the positions of the paths at time $t\neq t^{*}$ were shown to be
the usual scaling limits from random matrix theory. Here, we describe the limit
as $n\to \infty$ of the correlation kernel at critical time $t^{*}$ and in the
double scaling regime. We derive an integral representation for the limit
kernel which bears some connections with the Pearcey kernel. The analysis is
based on the study of a $3\times 3$ matrix valued Riemann-Hilbert problem by
the Deift-Zhou steepest descent method. The main ingredient is the construction
of a local parametrix at the origin, out of the solutions of a particular
third-order linear differential equation, and its matching with a global
parametrix.
