  In this paper we compute the intersection number of two double ramification
cycles (with different ramification profiles) and the top Chern class of the
Hodge bundle on the moduli space of stable curves of any genus. These quadratic
double ramification integrals are the main ingredient for the computation of
the double ramification hierarchy associated to the infinite dimensional
partial cohomological field theory given by $\exp(\mu^2 \Theta)$ where $\mu$ is
a parameter and $\Theta$ is Hain's theta class, appearing in Hain's formula for
the double ramification cycle on the moduli space of curves of compact type.
This infinite rank double ramification hierarchy can be seen as a rank $1$
integrable system in two space and one time dimensions. We prove that it
coincides with a natural analogue of the KdV hierarchy on a noncommutative
Moyal torus.
