Preferred instantaneous vacuum for linear scalar fields in cosmological space-times

We discuss the problem of defining a preferred vacuum state at a given time for a quantized scalar field in Friedmann, Lemaître, Robertson, Walker space-time. Among the infinitely many homogeneous, isotropic vacua available in the theory, we show that there exists at most one for which every Fourier mode makes a vanishing contribution to the adiabatically renormalized energy-momentum tensor at any given instant. For massive fields such a state exists in the most commonly used backgrounds in cosmology and, within the adiabatic regularization scheme, provides a natural candidate for the “ground state” at that instant of time. The extension to the massless and the conformally coupled case are also discussed.

Figure 1

FLRW space-time sourced by radiation: The left panel shows the behavior of ${\mathrm{\Omega }}_k(\eta )$, defined in Eq. (3.2) as a function of $\eta$ and $k$ and the right panel, the behavior of $r_k(\eta )$ defined in Eq. (3.5). Here we have set $m=10^{-5}$ in the natural Planck units. In this model the scale factor has the form $a(\eta )=a_0\eta$ whence the singularity occurs at $\eta =0$. For the range of parameters considered, both functions remain finite and positive except very near the big bang, satisfying the necessary and sufficient condition (3.4) and (3.5) of the existence of the instantaneous vacuum at time $\eta$.

Figure 2

FLRW space-time sourced by nonrelativistic matter: As in Fig 1, the left panel shows the behavior of ${\mathrm{\Omega }}_k(\eta )$ [defined in (3.2)] and the right panel, the behavior of $r_k(\eta )$ defined in (3.5). The mass parameter is again $m=10^{-5}$ in the natural Planck units. Again, except very near the big bang singularity ($\eta =0$), both functions remain finite and positive. Thus for the range of $\eta ,k$ shown, the existence of the instantaneous vacuum $|0_{{\eta }_0}⟩$ is ensured.

Figure 3

De Sitter space-time: Again, the left panel shows the behavior of ${\mathrm{\Omega }}_k(\eta )$ and the right panel, the behavior of $r_k(\eta )$. In this plot we have set $m=H=10^{-5}$ in the natural Planck units. Now, $\eta =0$ corresponds to future infinity which is not part of space-time. For the values of $\eta ,k$ considered, the functions remain finite and positive, ensuring the existence of the instantaneous vacuum $|0_{{\eta }_0}⟩$.

Figure 4

Growth of $r_k(\eta )$ in de Sitter (left) and radiation filled FLRW (right) space-times: This figure zooms in on the region in which $r_k(\eta )$ starts growing as one approaches $\eta =0$. The growth goes as $a^{\prime }/a\sim 1/{\eta }^2$. Behavior is similar for the matter-filled FLRW universe.