In a unified picture both inflation and present dynamical dark energy arise from the same scalar field .
The history of the Universe describes a crossover from a scale invariant “ past fixed point ” where all particles are massless , to a “ future fixed point ” for which spontaneous breaking of the exact scale symmetry generates the particle masses .
The cosmological solution can be extrapolated to the infinite past in physical time - the universe has no beginning .
This is seen most easily in a frame where particle masses and the Planck mass are field-dependent and increase with time .
In this “ freeze frame ” the Universe shrinks and heats up during radiation and matter domination .
In the equivalent , but singular Einstein frame cosmic history finds the familiar big bang description .
The vicinity of the past fixed point corresponds to inflation .
It ends at a first stage of the crossover .
A simple model with no more free parameters than \Lambda CDM predicts for the primordial fluctuations a relation between the tensor amplitude r and the spectral index n,r = 8.19 ( 1 - n ) -0.137 .
The crossover is completed by a second stage where the beyond-standard-model sector undergoes the transition to the future fixed point .
The resulting increase of neutrino masses stops a cosmological scaling solution , relating the present dark energy density to the present neutrino mass .
At present our simple model seems compatible with all observational tests .
We discuss how the fixed points can be rooted within quantum gravity in a crossover between ultraviolet and infrared fixed points .
Then quantum properties of gravity could be tested both by very early and late cosmology .