# Definition of vacuum and occupation number in expanding Universe

Suppose for simplicity we have theory of free quantum scalar field in expanding Universe (metric plays the role of background field) $g_{\mu \nu} = \text{diag}(1, -a^2,-a^2,-a^2)$, where $a(t) \sim \sqrt{t}$ as for radiation dominated universe: $$S = \int d^{4}x\sqrt{-g}\left( \frac{1}{2}g^{\mu \nu}\partial_{\mu}\theta \partial_{\nu}\theta - \frac{1}{2}m^2(t)\theta^2\right)$$ Here time dependence of mass $m(t)$ is such that mass is adiabatically changing quantity: $$\left|\frac{\dot{m}(t)}{m(t)}\right|<m(t)$$ By expanding $\theta$ in a series $$\theta = \frac{1}{\sqrt{V(t)}}\sum_{k}e^{i\mathbf k \cdot \mathbf x}\left( \hat{a}_{k}f_{k}(t) + \hat{a}^{\dagger}_{-k}f_{k}(t)^{*}\right),$$ where the comoving volume is $V \sim a(t)^3$, and deriving EOM for modes $f_{k}$ we obtain $$\tag 1 \ddot{f}_{k}(t) + \left( \frac{k^2}{a^2} + m^2(t) - \frac{\ddot{a}}{a}\right)f_{k}(t) = 0$$ Eq. $(1)$ has solution in terms of Bessel functions. For case of infinite large time, $t \to \infty$, it has asymptotic expansion in a plane waves. But for case of zero time it hasn't. The reason is that for $t \to 0$ (or to some $t \to t_{0}$, where $t_{0}$ is very small) all modes are outside the horizon, i.e. there also are moments of time $t$, for each for given wavenumber $k$ following inequality holds: $k < a(t)H(t)$. Thus it is hard to determine what is vacuum in theory, and hence, creation of particles due to time dependence of $m(t)$.

How people solve this problem?

• The vacuum in QFTs on curved (non-Minkowski) is not unique, as you just found it. There are several different definitions of "vacua" floating around which are all appropriate for different computations, but in general, this problem isn't so much "solved" as "lived with". – ACuriousMind Dec 7 '15 at 16:07
• @ACuriousMind : but people say that, for example, coherent oscillations of scalar field in an expanding Universe produces particles. This statement requires precise definition of particle and vacuum, doesn't it? – Name YYY Dec 7 '15 at 16:15
• Yes, and the people who say that should somewhere mention what their notion of vacuum is. – ACuriousMind Dec 7 '15 at 16:17