Red or blue tensor spectrum from GW170817-compatible Einstein-Gauss-Bonnet theory: A detailed analysis

Abstract In this work we shall prove that the tensor spectral index of the
primordial tensor perturbations for GW170817-compatible
Einstein-Gauss-Bonnet theories, takes the approximate simplified
form $n_{\mathcal{T}}\simeq 2\left(-1+\frac{1}{\lambda(\phi)}
\right)\epsilon_1$ at leading order, with $\lambda (\phi)$ being a
function of the scalar field which depends on the scalar field
potential and the second derivative of the scalar-Gauss-Bonnet
coupling $\xi''(\phi)$. With our analysis we aim to provide a
definitive criterion for selecting Einstein-Gauss-Bonnet models
that can provide a blue-tilted inflationary phenomenology, by
simply looking at the scalar potential and the scalar-Gauss-Bonnet
coupling. We shall prove this using two distinct approaches and as
we show the tilt of the tensor spectral index is determined by the
values of the potential $V(\phi)$ and of scalar-Gauss-Bonnet
coupling at first horizon crossing. Specifically the blue-tilted
tensor spectral index can occur when $\xi''(\phi_*)V(\phi_*)>0$ at
first horizon crossing.