  We consider the properties of stress-energy tensors compatible with a Null
Big Bang, i.e., cosmological evolution starting from a Killing horizon rather
than a singularity. For Kantowski-Sachs cosmologies, it is shown that if matter
satisfies the Null Energy Condition (NEC), then (i) regular cosmological
evolution can only start from a Killing horizon, (ii) matter is absent at the
horizon, and (iii) matter can only appear in the cosmological region due to
interaction with vacuum. The latter is understood phenomenologically as a fluid
whose stress tensor is insensitive to boosts in a particular direction. We also
argue that matter is absent in a static region beyond the horizon. All this
generalizes the observations recently obtained for a mixture of dust and a
vacuum fluid. If, however, we admit the existence of phantom matter, its
certain special kinds (with the parameter $w \leq -3$) are consistent with a
Null Big Bang without interaction with vacuum (or without vacuum fluid at all).
Then in the static region there is matter with $w\geq -1/3$. Alternatively, the
evolution can begin from a horizon in an infinitely remote past, leading to a
scenario combining the features of a Null Big Bang and an emergent universe.
