Exact hyperbolic-sine scalar-field cosmology
= Exact hyperbolic-sine scalar-field cosmology
In a flat universe containing a homogeneous canonical <scalar field>, the ansatz
$$
\phi(t)=\phi_0\log\tanh(\lambda t)
$$
has the exact solution
$$
H=\lambda\frac{\phi_0^2}{M_{\rm Pl}^2}\coth(2\lambda t),
\qquad
a=a_0[\sinh(2\lambda t)]^{\phi_0^2/(2M_{\rm Pl}^2)}.
$$
It interpolates between decelerating power-law expansion at early times and exponential accelerated expansion at late times when $2/3<\phi_0^2/M_{\rm Pl}^2<2$.