Uncertainty in mode number operator and Hawking's original paper

I began reading Hawking's paper Particle Creation by Black Holes (1975, Commun. math. Phys 43, 199—220) but am a little confused by what he writes at the bottom of the second page. The idea is that there is some indeterminacy or uncertainty in the mode number operator $$a_i a_i^\dagger$$ in curved spacetime.

What Hawking does: He first goes into Riemann normal coordinates which are valid up to some length scalar, say $$\ell$$. In Hawking's language $$\ell=B^{-1/2}$$ where $$B$$ is a least upper bound on $$|R_{abcd}|$$, so $$\ell$$ is a radius of curvature and the flat space limit is given by $$\ell\rightarrow \infty$$. Next, since this is locally flat space, he is allowed to choose a basis of (approximately) positive frequency solutions to the wave equation, $$\{f_i\}$$. Finally, he writes that there is an indeterminacy between choosing $$f_i$$ and its corresponding negative frequency solution $$f_i^*$$ which is of the order $$\exp(-c \omega \ell)$$. Here I have let $$c$$ be some constant, and $$\omega$$ is the (modulus) frequency of the mode in question.

My Question: I have a hard time understanding what he means by this final part. What does he mean precisely by 'indeterminacy'? Why is there an exponential involved?

My Intuition: I have the following picture: it follows from the Heisenberg uncertainty principle that $$\Delta E \Delta t \sim 1$$. In units where $$\hbar =1$$ one has uncertainty $$\Delta \omega = \Delta E \sim 1/\Delta t$$. Since $$\Delta t$$ is bounded by $$B^{-1/2}$$ in the normal coordinates, we have a minimal uncertainty in frequency of order $$\Delta \omega \sim B^{1/2}$$.

So we can imagine two normal distributions, one for $$f_i$$ and one for $$f_i^*$$, centered at $$\pm\omega$$, each having standard deviation $$B^{1/2}$$.

There are two extreme cases:

1. When $$\omega \gg B^{1/2}$$, the two normal distributions are far apart and one is exponentially sure that a mode which is measured to have positive frequency really is a positive frequency mode.

2. When the two distributions are close, i.e. when $$\omega \lesssim B^{1/2}$$, one might expect increasingly equal probabilities (close to $$1/2$$).

In the former case one can use an asymptotic of the normal distribution to show that the probability of a negative frequency mode to be measured as positive is of order $$\sim \frac{1}{2\sqrt{\pi} \alpha}e^{-\alpha^2}$$ where $$\alpha=\frac{1}{\sqrt{2}}\omega B^{-1/2}$$. Whilst qualitatively this is the same as Hawking's result, it differs quantitatively - I have an $$\alpha^2$$ in the exponent, whilst Hawking only has $$\alpha$$. What am I doing wrong, and what is Hawking doing?!

A bonus question: Does anyone know / can anyone give a more rigorous derivation of the uncertainty in the mode number?

Many thanks.