In search of an observational quantum signature of the primordial perturbations in slow-roll and ultraslow-roll inflation

In the standard inflationary paradigm, cosmological density perturbations are generated as quantum fluctuations in the early Universe, but then undergo a quantum-to-classical transition. A key role in this transition is played by squeezing of the quantum state, which is a result of the strong suppression of the decaying mode component of the perturbations. Motivated by ever improving measurements of the cosmological perturbations, we ask whether there are scenarios where this decaying mode is nevertheless still observable in the late Universe, ideally leading to a “smoking gun” signature of the quantum nature of the perturbations. We address this question by evolving the quantum state of the perturbations from inflation into the postinflationary Universe. After recovering the standard result that in slow-roll (SR) inflation the decaying mode is indeed hopelessly suppressed by the time the perturbations are observed (by $\sim 115$ orders of magnitude), we turn to ultraslow-roll (USR) inflation, a scenario in which the usual decaying mode actually grows on superhorizon scales. Despite this drastic difference in the behavior of the mode functions, we find also in USR that the late-Universe decaying mode amplitude is dramatically suppressed, in fact by the same $\sim 115$ orders of magnitude. We finally explain that this large suppression is a general result that holds beyond the SR and USR scenarios considered and follows from a modified version of Heisenberg’s uncertainty principle and the observed amplitude of the primordial power spectrum. The classical behavior of the perturbations is thus closely related to the classical behavior of macroscopic objects drawing an analogy with the position of a massive particle, the curvature perturbations today have an enormous effective mass of order $m_{\mathrm{pl}}^2/H_0^2\sim {10}^{120}$, making them highly classical.

Figure 1

Left: The growing mode (${\mathcal{R}}_{\mathrm{grow}}^{\mathrm{SR}}$, red) and decaying mode (${\mathcal{R}}_{\mathrm{dec}}^{\mathrm{SR}}$, blue) characterizing the quantum state of the primordial perturbations in slow-roll inflation, as a function of the number of $e$-foldings of expansion since horizon exit, $N$. As a mode exits the horizon (at $N=0$), the growing mode approaches a constant, while the decaying mode tends to zero. We have normalized the modes such that ${\mathcal{R}}_{\mathrm{grow}}^{\mathrm{SR}}\to 1$ in the superhorizon limit. Right: The rapid suppression (proportional to $\sim e^{-3N}$, where $N$ is the number of $e$-foldings since horizon exit) of the decaying mode relative to the growing mode leads to a highly squeezed state, characterized by a narrow Wigner function. We show (in arbitrary units) Wigner contours of constant ${\chi }^2\equiv -2\ln (\pi W(\mathcal{R},\mathrm{\Pi }))$ for various values of ${\ell }_k/{\ell }_H=e^N$, the ratio of the mode wavelength to the Hubble scale (see inset). The highly squeezed superhorizon state behaves classically in the way discussed in the text.

Figure 2

A Wigner ellipse (${\chi }^2=1$, see text) in the squeezed limit, in terms of $\mathcal{R}$ and ${\mathcal{R}}^{\prime }$. The extent in the $\mathcal{R}$ and ${\mathcal{R}}^{\prime }$ directions gives the root-mean-square values of these operators (just like ${\chi }^2=1$ contours of a true phase-space distribution would). In the squeezed limit, the extended direction of the ellipse (dashed line) is described by the growing mode and the growing mode components of the operators $\widehat{\mathcal{R}}$ and ${\widehat{\mathcal{R}}}^{\prime }$ commute. We thus colloquially refer to the growing mode contribution as the classical component. The decaying mode describes the nonzero width of the ellipse. Since it is the inclusion of the decaying mode that causes the noncommutation of $\widehat{\mathcal{R}}$ and ${\widehat{\mathcal{R}}}^{\prime }$, we will loosely refer to the decaying mode as the quantum contribution. With this convention, the quantum spreads in $\mathcal{R}$ and ${\mathcal{R}}^{\prime }$, $\delta {\mathcal{R}}_{\mathrm{qu}}$ and $\delta {\mathcal{R}}_{\mathrm{qu}}^{\prime }$ are as indicated in the figure and become small relative to the classical spread as the decaying mode gets more and more suppressed and the Wigner ellipse narrower. We stress that the primordial state under consideration is fully quantum mechanical even in the squeezed limit. We here use the labels quantum and classical only in the specific sense explained above and in the text.

Figure 3

Ratio of the wavelength of a mode with wave number $k$ to the Hubble scale, as a function of the number of $e$-foldings of expansion since horizon exit. There are $N_{*}$ $e$-foldings between horizon exit and the end of inflation. We then assume an instantaneous transition to a radiation dominated hot big bang phase. In a simplified Universe where the Universe remains radiation dominated (and the number of radiation degrees of freedom is constant), it takes exactly another $N_{*}$ $e$-foldings before the mode $k$ reenters the horizon.

Figure 4

Evolution of the growing and decaying modes describing the primordial quantum state in a simplified version of the slow-roll scenario, where the postinflationary phase is radiation dominated even at late times. Left: Evolution of the normalized growing mode $a_{\mathcal{R}}^{-1}{\mathcal{R}}_1$ (red) and decaying mode $a_{\mathcal{R}}^{-1}{\mathcal{R}}_2$ (blue) a few $e$-foldings before and after horizon exit. Center: Evolution in the superhorizon regime. The decaying mode is suppressed by a factor $e^{-3N_{*}}$ during inflation and the basis rotation associated with the transition to the RD phase (see text) accounts for an additional suppression of order ${\epsilon }_{*}$. During RD, the decaying mode is suppressed by another factor $e^{-N_{*}}$. Right: Evolution a few $e$-foldings before and after horizon reentry during RD. The decaying mode is rescaled in order to make it visible. In reality it is down by the cumulative suppression factor $\sim {\epsilon }_{*}{e}^{-4N_{*}}\lesssim {10}^{-96}$ (for $N_{*}\approx 55$). This reflects the standard result that (ignoring decoherence) the state of the perturbations is extremely squeezed and thus classical in the sense described in the text.

Figure 5

Illustration of effect of the decaying mode on the dimensionless power spectrum of curvature perturbations. We assume a toy model where the late Universe is dominated by a perfect radiation fluid (so that we can use analytic solutions), and the power spectrum is evaluated at a time where the sound horizon $s_{\mathrm{obs}}=c_s^{\mathrm{RD}}{\tau }_{\mathrm{obs}}=100h^{-1}\mathrm{Mpc}$, in order to mimic the acoustic oscillations of the real Universe. We consider a nearly flat primordial spectrum for the growing mode ($n_s=0.96$) and the decaying mode amplitude is assigned an arbitrary amplitude [cf. Eqs (58) and (60)]. The spectrum is shown for decaying mode amplitudes ${\mathrm{\Delta }}_{\mathrm{dec},p}=0$ (black), ${10}^{-4}$ (red dashed) and $3\times {10}^{-4}$ (blue dashed). The decaying mode, which encodes the noncommuting component of the operators describing the primordial perturbations, leads to a scale-dependent damping of the acoustic oscillations.

Figure 6

Superhorizon evolution of growing and decaying modes during USR inflation ($N<N_{*}$) and radiation domination ($N>N_{*}$). In the USR scenario, the growing and decaying modes switch roles compared to standard slow-roll inflation: the nonconstant mode that would be a decaying mode in SR, ${\mathcal{R}}_2\sim {\mathcal{R}}_{\mathrm{dec}}^{\mathrm{USR}}$, grows rapidly in the superhorizon regime, whereas the usual growing mode, ${\mathcal{R}}_1\sim {\mathcal{R}}_{\mathrm{grow}}^{\mathrm{USR}}(\tau )$, is constant as in SR and evolves into the RD constant mode ${\mathcal{R}}_{\mathrm{grow}}^{\mathrm{RD}}$. The question is then what does the mode ${\mathcal{R}}_{\mathrm{dec}}^{\mathrm{USR}}$ evolve into in the RD regime? If it matches onto the RD decaying mode, as hinted at above (but note the question mark), one could imagine late-Universe scenarios where the RD decaying mode is not or less suppressed and the state is not fully classical, perhaps leaving an interesting observable signature. We address whether this is the case in Sec. 4 (and Fig. 7).

Figure 7

As Fig. 4, but with the inflationary phase governed by the USR scenario. Explicit calculation of the evolution of the inflationary mode ${\mathcal{R}}_2$ into the RD regime shows that it evolves mostly into the RD growing mode (unlike the hypothetical scenario depicted with the question mark in Fig. 6). Expanding in terms of growing and decaying modes with respect to late-Universe behavior, we find that at the start of the RD phase, the RD decaying mode is suppressed by a factor ${\epsilon }_{*}{e}^{-9N_{*}}$, of which $e^{-3N_{*}}$ is due to the growth of ${\mathcal{R}}_2$ during USR inflation, and the remaining ${\epsilon }_{*}{e}^{-6N_{*}}$ due to the basis transformation associated with the transition at the end of inflation, leading the mode ${\mathcal{R}}_{1^{\prime }}$ to evolve into the RD decaying mode. As in the standard scenario, superhorizon evolution after inflation contributes another $e^{-N_{*}}$ suppression. Thus, even in the USR scenario, the resulting state of the perturbations is extremely squeezed and quantum effects of the type discussed in the text are hopelessly suppressed. We explain in Sec. 5 that, in single-field inflation, this will in fact be the case no matter what exotic scenario we consider to modify the behavior of the growing and decaying modes.