  We calculate the power spectrum, spectral index, and running spectral index
for inflationary patch cosmology arisen from Gauss-Bonnet braneworld scenario
using the Mukhanov equation. This patch cosmology consists of Gauss-Bonnet(GB),
Randall-Sundrum (RS-II), and four dimensional (4D) cosmological models. There
exist several modifications in higher order calculations. However, taking the
power-law inflation by choosing different potentials depending on the model,
there exist minor changes up to second order corrections. Since second order
corrections are rather small in the slow-roll limit, we could not choose a
desired power-law model which explains the WMAP data. Finally we discuss the
reliability of high order calculations based on the Mukhanov equation by
comparing the perturbed equation including 5D metric perturbations. It turns
out that first order corrections are reliable, while second order corrections
are not proved to be reliable.
