I will argue that discreteness at the Planck scale (naturally expected to arise from quantum gravity) might manifest in the form of minute violations of energy-momentum conservation of the matter degrees of freedom when described in terms of (idealized) smooth fields on a smooth spacetime. In the context of applications to cosmology such `energy diffusion' from the low energy matter degrees of freedom to the discrete structures underlying spacetime leads to the emergence of an effective dark energy term in Einstein's equations.

The interest in primordial black holes (PBHs) has recently risen up 
again after the discovery of around 30 solar mass black holes through 
the LIGO/Virgo gravitational wave events. PBHs are black holes produced 
by gravitational collapse of the over-dense region in the early universe.
 An attractive origin of the overdensity is primordial perturbations 
generated during inflation. In this talk, I will first explain the 
relation between the power spectrum of the perturbations predicted by 

When a Hamiltonian density is bounded by below, we know that
the lowest-energy state must be stable. One is often tempted
to reverse the theorem and therefore believe that an unbounded
Hamiltonian density always implies an instability. The main
purpose of this talk is to pedagogically explain why this is
erroneous. Stability is indeed a coordinate-independent property,
whereas the Hamiltonian density does depend on the choice of
coordinates. In alternative theories of gravity, like k-essence
or Horndeski theories, the correct stability criterion is