Semiclassical predictions regarding a preinflationary era and its effects on the power spectrum

An investigation is undertaken into the properties and effects of a preinflationary era during at least part of which semiclassical gravity was valid. It is argued that if the Universe (or our part of it) was approximately homogeneous and isotropic during that era, then the Universe was likely to have been radiation dominated. A simple model in which the Universe contains classical radiation and a cosmological constant is used to investigate potential effects of such a preinflationary era on the cosmic microwave background. The power spectrum is computed using the mode functions of a quantized massless minimally coupled scalar field. Various choices of state for this field are considered, including adiabatic vacuum states of various orders and the vacuum state that would naturally occur if the Universe made a sudden transition from being radiation dominated to de Sitter space. In all cases investigated, there is a suppression of the power spectrum at large angles, and, when plotted as a function of the momentum parameter, there are always oscillations with state-dependent amplitudes.

Figure 1

Rescaled scale factor $\alpha (\chi )$ over its domain $({\chi }_0,{\chi }_{\infty })$.

Figure 2

Power spectrum from a sudden approximation with the transition occurring at ${\chi }_s=0$. Enhancements in the power spectrum are seen compared to the Bunch-Davies value, represented by the dotted line.

Figure 3

Comparison of the power spectra for adiabatic vacuum states of order zero, two, and four, when the matching is done at ${\alpha }_m=0.1$ (${\chi }_m=-0.827$). Note that all orders show oscillatory behavior, but this behavior is much smaller at higher orders. Note also that the power spectrum never noticeably exceeds 1, the Bunch-Davies value.

Figure 4

Comparison of the power spectra for adiabatic vacuum states of order zero, two, and four, when the matching is done at ${\alpha }_m=1$ (${\chi }_m=0$).

Figure 5

Contributions to the angular power spectrum from modes with $\kappa \ge \sqrt{2}{\alpha }_m$ for a fourth-order adiabatic vacuum state with the adiabatic matching done at ${\alpha }_m=0.1$. From top to bottom, the curves are for $s=0.50$, 0.30, 0.20, 0.10, and 0.05.

Figure 6

Angular power spectrum for the sudden approximation and for the contributions from modes with $\kappa \ge \sqrt{2}{\alpha }_m$ to the power spectrum for zeroth- and fourth-order adiabatic vacuum states. For the sudden approximation the matching is done at ${\alpha }_s=1$, while for the adiabatic vacuum states the matching was done at ${\alpha }_m=0.1$. In each case, $s=0.3$.