The inflaton field exists in a superposition of energies. How does that affect the rate of cosmological expansion? The inflaton field is a scalar field that each field value corresponds to different rates of expansion.
The field exists in a superposition of energies, that is, the wave function hasn’t collapsed. Like other quantum field, the field value at each point is described as a probability distribution over a range of possible values.  Since the field value is fuzzy/uncertain/not precisely determined, how would the expansion rate at each point in space be?
Would the expansion rate at a point in space at an instant in time be precisely the average value, or fluctuate randomly around the average value, or vary from point to point across space around the possible value, or simply cannot be precisely determined?
 A: Let's start with what is definitely known. Suppose you place an atom in a superposition of the ground state $|g\rangle$ and an excited state $|e\rangle$ in vacuum, and wait for much longer than the decay time. What's the state of the electromagnetic field afterward? 
We know that $|g \rangle$ will stay in the ground state, so the electromagnetic field will remain in the vacuum state. But $|e \rangle$ will decay to $|g \rangle$, putting one photon into the electromagnetic field. So the actual state of the field would be a superposition of these two. You can convert this into a probabilistic mixture (i.e. you either get no photons, or some photons) by measuring the field. This is the standard way to treat quantum objects interacting with fields, and it is textbook material that is extremely well-tested.
The same is done in inflation. Quantum objects like the inflaton are coupled to the metric. The metric field is measured (depending on who you ask, this occurs effectively by decoherence processes in the early universe, or by us ourselves when we observe the universe), and hence becomes a probabilistic mixture. This general idea is known as stochastic inflation. For an early reference, see the seminal paper Stochastic de Sitter (Inflationary) Stage in the Early Universe (1986) by Starobinsky.
For some inflation models, the fluctuations of the inflaton field are small, so the expansion rate is precisely determined. When the fluctuations are large, you get quite a wide range of expansion rates, and as a result inflation is generically eternal, leading to difficulties with calculating probabilities. 
Obviously, assumptions about quantum gravity have been made here. However, an assumption of some sort is absolutely necessary to make progress, the particular assumptions made are reasonable and don't require knowledge of Planck-scale physics, and the resulting predictions of, e.g. perturbations of the CMB have been well-confirmed.
