Mechanism for the suppression of quantum noise at large scales on expanding space

We present an exactly solvable model for the suppression of quantum noise at large scales on expanding space. The suppression arises naturally in the de Broglie-Bohm pilot-wave formulation of quantum theory, according to which the Born probability rule has a dynamical origin. For a scalar field on a radiation-dominated background we construct the exact solution for the time-evolving wave functional and study properties of the associated field trajectories. It is shown that the time evolution of a field mode on expanding space is mathematically equivalent to that of a standard harmonic oscillator with a “retarded time” that depends on the wavelength of the mode. In the far super-Hubble regime the equivalent oscillator evolves over only one Hubble time, yielding a simple mechanism whereby relaxation to the Born rule can be suppressed on very large scales. We present numerical simulations illustrating how the expansion of space can cause a retardation of relaxation in the super-Hubble regime. Given these results it is natural to expect a suppression of quantum noise at super-Hubble wavelengths. Such suppression could have taken place in a preinflationary era, resulting in a large-scale power deficit in the cosmic microwave background.

Figure 1

Plot of the retarded time $t_{\mathrm{ret}}=t_{\mathrm{ret}}(t)$ (solid line) for $t$ on the interval $(t_i,t_{\mathrm{enter}})$. The dotted line is a plot of real time $t$. The function $t_{\mathrm{ret}}(t)$ is given in terms of $\Theta (t)$ by $t_{\mathrm{ret}}(t)=t_i+\Theta (t)/{\omega }_i$ where $\Theta (t)$ is given by Eq. (37). We have chosen parameters $t_i={10}^{-4}$, $a_i={10}^{-2}$ and $k=10\pi$ (so that $\epsilon =100{\pi }^2$).

Figure 2

Time evolution of nonequilibrium on expanding space, for a superposition of 25 modes. Results for the interval $(t_i,t_{\mathrm{enter}})$ are obtained by evolving the equivalent oscillator over the retarded interval $(t_i,t_{\mathrm{ret}}(t_{\mathrm{enter}}))$. The (smoothed) actual distribution ${\tilde{\rho }}^{\prime }$ is displayed in the left column, the (smoothed) equilibrium distribution ${\tilde{\rho }}_{\mathrm{QT}}^{\prime }$ in the right column. The top row shows the distributions at the initial time $t_i$, the second row at an intermediate retarded time $t_{\mathrm{ret}}(0.5t_{\mathrm{enter}})$, and the third row at $t_{\mathrm{ret}}(t_{\mathrm{enter}})$. The support of ${\tilde{\rho }}^{\prime }$ remains significantly narrower than the support of ${\tilde{\rho }}_{\mathrm{QT}}^{\prime }$.

Figure 3

Time evolution of the same initial state as in Figure 2 but with no spatial expansion. The results for $(t_i,t_{\mathrm{enter}})$ are now obtained simply by evolving the standard oscillator over $(t_i,t_{\mathrm{enter}})$. The top row again shows the (smoothed) distributions at the initial time $t_i$, the second row at the intermediate time $0.5t_{\mathrm{enter}}$, and the third row at $t_{\mathrm{enter}}$. Already at $t=0.5t_{\mathrm{enter}}$ the actual distribution ${\tilde{\rho }}^{\prime }$ has a support that is only slightly narrower than the support of ${\tilde{\rho }}_{\mathrm{QT}}^{\prime }$. At $t=t_{\mathrm{enter}}$ there is little discernible difference between ${\tilde{\rho }}^{\prime }$ and ${\tilde{\rho }}_{\mathrm{QT}}^{\prime }$—relaxation is almost complete.

Figure 4

Plots of $\ln \overline{H}$ against time $t$, with spatial expansion (upper curve) and with no spatial expansion (lower curve). Real time runs from $t=t_i$ up to $t=t_{\mathrm{enter}}$. The lower curve has a larger (negative) slope and ends with a smaller value. With no spatial expansion there is faster relaxation and the final distribution comes considerably closer to equilibrium. The difference between the two $\overline{H}$-curves quantifies the suppression of relaxation on expanding space in the super-Hubble regime.

Figure 5

Inflation with a radiation-dominated preinflationary era. The dashed line shows the Hubble radius $H^{-1}$. The solid lines show physical wavelength ${\lambda }_{\mathrm{phys}}$ for two different modes: the lower line enters the Hubble radius during preinflation and exits during inflation, while the upper line remains outside the Hubble radius throughout the preinflationary era and enters only during the transition.