  We further develop perturbative methods used to calculate entanglement
entropy (EE) away from an interacting CFT fixed point. At second order we find
certain universal terms in the renormalized EE which were predicted previously
from holography and which we find hold universally for relevant deformations of
any CFT in any dimension. We use both replica methods and direct methods to
calculate the EE and in both cases find a non-local integral expression
involving the CFT two point function. We show that this integral expression can
be written as a local integral over a higher dimensional \emph{bulk} modular
hamiltonian in an emergent $AdS$ space-time. This bulk modular hamiltonian is
associated to an emergent scalar field dual to the deforming operator. We
generalize to arbitrary spatially dependent couplings where a linearized metric
emerges naturally as a way of efficiently encoding the field theory
entanglement: by demanding that Einstein's equations coupled to the bulk scalar
field are satisfied, we show that EE can be calculated as the area of this
metric. Not only does this show a direct emergence of a higher dimensional
gravitational theory from any CFT, it allows for effective evaluation of the
the integrals required to calculate EE perturbativly. Our results can also be
interpreted as relating the non-locality of the modular hamiltonian for a
spherical region in non-CFTs and the non-locality of the holographic bulk to
boundary map.
