  We investigate the effect of non-linear dispersion relations on the
bispectrum. In particular, we study the case were the modified relations do not
violate the WKB condition at early times, focusing on a particular example
which is exactly solvable: the Jacobson-Corley dispersion relation with quartic
correction with positive coefficient to the squared linear relation. We find
that the corrections to the standard result for the bispectrum are suppressed
by a factor $\frac{H^2}{p_c^2}$ where $p_c$ is the scale where the modification
to the dispersion relation becomes relevant. The modification is {\it mildly}
configuration-dependent and equilateral configurations are more suppressed with
respect to the local ones, by a factor of one percent. There is no
configuration leading to enhancements. We then analyze the results in the
framework of particle creation using the approximate gluing method of
Brandenberger and Martin, which relates more directly to the modeling of the
trans-Planckian physics via modifications of the vacuum at a certain cutoff
scale. We show that the gluing method overestimates the leading order
correction to the spectrum and bispectrum by one and two orders, respectively,
in $\frac{H}{p_c}$. We discuss the various approximation and conclude that for
dispersion relations not violating WKB at early times the particle creation is
small and does not lead to enhanced contributions to the bispectrum. We also
show that in many cases enhancements do not occur when modeling the
trans-Planckian physics via modifications of the vacuum at a certain cutoff
scale. Most notably they are only of order O(1) when the Bogolyubov
coefficients accounting for particle creation are determined by the Wronskian
condition and the minimization of the uncertainty between the field and its
conjugate momentum.

High Energy Physics - Theory;Cosmology and Nongalactic Astrophysics;General Relativity and Quantum Cosmology;High Energy Physics - Phenomenology