The Einstein-Gauss-Bonnet equations projected from the bulk to brane lead to a complicated Friedmann equation which simplifies to H ^ { 2 } \sim \rho ^ { q } in the asymptotic regimes .
The Randall-Sundrum ( RS ) scenario corresponds to q = 2 whereas q = 2 / 3 \& q = 1 give rise to high energy Gauss-Bonnet ( GB ) regime and the standard GR respectively .
Amazingly , while evolving from RS regime to high energy GB limit , one passes through a GR like region which has important implications for brane world inflation .
For tachyon GB inflation with potentials V ( \phi ) \sim \phi ^ { p } investigated in this paper , the scalar to tensor ratio of perturbations R is maximum around the RS region and is generally suppressed in the high energy regime for the positive values of p .
The ratio is very low for p > 0 at all energy scales relative to GB inflation with ordinary scalar field .
The models based upon tachyon inflation with polynomial type of potentials with generic positive values of p turn out to be in the 1 \sigma observational contour bound at all energy scales varying from GR to high energy GB limit .
The spectral index n _ { S } improves for the lower values of p and approaches its scale invariant limit for p = -2 in the high energy GB regime .
The ratio R also remains small for large negative values of p , however , difference arises for models close to scale invariance limit .
In this case , the tensor to scale ratio is large in the GB regime whereas it is suppressed in the intermediate region between RS and GB .
Within the frame work of patch cosmologies governed by H ^ { 2 } \sim \rho ^ { q } , the behavior of ordinary scalar field near cosmological singularity and the nature of scaling solutions are distinguished for the values of q < 1 and q > 1 .
The tachyon dynamics , on the other hand , exhibits stable scaling solutions \forall q if the adiabatic index of barotropic fluid \gamma < 1 .