How do Graviton-based theories of gravity explain the expansion of the universe? In General Relativity, the expansion of the universe is modeled using the Friedmann–Lemaître–Robertson–Walker metric, and the expansion itself is a metric expansion by which the scale of space itself changes over time. Using the Einstein field equations you can derive the Friedmann equations then solve for the scale factor, $a(t)$. 
However in recent history there have be many attempts to model a quantum theory of gravity in terms of a graviton which mediates the force of gravity, often in background-dependent theories such as string theory.
My question is, how could a theory like this, especially a background dependant one, model the expansion of the universe? If gravity is modeled as a result of a spin-2 particle and not the curvature of spacetime, how can the metric expansion of space occur?  
Any answers, especially pertaining to Quantum Gravity as an Effective Field Theory, or String Theory, would be much appreciated. 
EDIT: I have found an article that states that gravitons are fluctuations of the geometry around flat space-time. Perhaps my mistake was assuming that gravitons and space-time curvature are mutually exclusive. As i do not have a great understanding of QFT, this would not be surprising.  
 A: This is not an exact duplicate of this previous question because it is asking about the connection with the expansion of the universe.
Already in the present model of the expansion of the universe an effective quantum mechanical theory is invoked in order to model the cosmic microwave background (CMB)observations and the horizon problem.. In the quantum mechanical model there are no gravitons, but inflatons that are homogenizing the very early universe to the level observed. This gives a brief but very large expansion of the unviverse at that time:

Graviton based theories of gravity are models relevant after the inflation period, when particles as we know them form from the primal energy. Gravitons have no role in the expanding of the universe at the rate of the simple Big Bang to which it returns after inflation. There are observations of an accelerated expansion rate, and there it is an open research problem, whether inflatons continue to have a role, or what model fits the observations.

My question is, how could a theory like this, especially a background dependant one, model the expansion of the universe? If gravity is modeled as a result of a spin-2 particle and not the curvature of spacetime, how can the metric expansion of space occur? 

One must not confuse frameworks in physics. For example, the complete thermodynamic theory framework  emerges from the underlying statistical mechanics , with completely different variables, but it has been shown that this emergence is compatible with the data.
There are only effective quantum mechanical models for quantization of gravity. The expectation is that the definitive model will reproduce the macroscopic frame of General Relativity . (otherwise it would not be a correct quantum mechanical underlying framework). Here is an answer on how general relativity can emerge from string theory, which has gravitons. All this is a matter of current research.
The  expansion  of the universe itself is an emergent effect using   the mathematical framework of General Relativity. The expectation is that this emergence will be proven rigorously once a definitive model of the quantization of gravity is found.
A: Gravitons are fluctuations of the gravitational field, just as photons are fluctuations of the electromagnetic field. Thus the field can still be warped in the GR sense. 
