  Using arguments built on ergodicity, we derive an analytical expression for
the Renyi entanglement entropies corresponding to the finite-energy density
eigenstates of chaotic many-body Hamiltonians. The expression is a universal
function of the density of states and is valid even when the subsystem is a
finite fraction of the total system - a regime in which the reduced density
matrix is not thermal. We find that in the thermodynamic limit, only the von
Neumann entropy density is independent of the subsystem to the total system
ratio $V_A/V$, while the Renyi entropy densities depend non-linearly on
$V_A/V$. Surprisingly, Renyi entropies $S_n$ for $n > 1$ are convex functions
of the subsystem size, with a volume law coefficient that depends on $V_A/V$,
and exceeds that of a thermal mixed state at the same energy density. We
provide two different arguments to support our results: the first one relies on
a many-body version of Berry's formula for chaotic quantum mechanical systems,
and is closely related to eigenstate thermalization hypothesis. The second
argument relies on the assumption that for a fixed energy in a subsystem, all
states in its complement allowed by the energy conservation are equally likely.
We perform Exact Diagonalization study on quantum spin-chain Hamiltonians to
test our analytical predictions, and find good agreement.
