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 $ellrightarrow infty$. 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 $pmomega$, 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 $sim frac{1}{2sqrt{pi} alpha}e^{-alpha^2}$ 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.