Cosmological inflation and the quantum measurement problem

According to cosmological inflation, the inhomogeneities in our Universe are of quantum-mechanical origin. This scenario is phenomenologically very appealing as it solves the puzzles of the standard hot big bang model and naturally explains why the spectrum of cosmological perturbations is almost scale invariant. It is also an ideal playground to discuss deep questions among which is the quantum measurement problem in a cosmological context. Although the large squeezing of the quantum state of the perturbations and the phenomenon of decoherence explain many aspects of the quantum-to-classical transition, it remains to understand how a specific outcome can be produced in the early Universe, in the absence of any observer. The continuous spontaneous localization (CSL) approach to quantum mechanics attempts to solve the quantum measurement question in a general context. In this framework, the wave function collapse is caused by adding new nonlinear and stochastic terms to the Schrödinger equation. In this paper, we apply this theory to inflation, which amounts to solving the CSL parametric oscillator case. We choose the wave function collapse to occur on an eigenstate of the Mukhanov-Sasaki variable and discuss the corresponding modified Schrödinger equation. Then, we compute the power spectrum of the perturbations and show that it acquires a universal shape with two branches, one which remains scale invariant and one with $n_{\mathrm{S}}=4$, a spectral index in obvious contradiction with the cosmic microwave background anisotropy observations. The requirement that the non-scale-invariant part be outside the observational window puts stringent constraints on the parameter controlling the deviations from ordinary quantum mechanics. Due to the absence of a CSL amplification mechanism in field theory, this also has the consequence that the collapse mechanism of the inflationary fluctuations is not efficient. Then, we determine the collapse time. On small scales the collapse is almost instantaneous, and we recover exactly the behavior of the CSL harmonic oscillator (a case for which we present new results), whereas, on large scales, we find that the collapse is delayed and can take several e-folds to happen. We conclude that recovering the observational successes of inflation and, at the same time, reaching a satisfactory resolution of the inflationary “macro-objectification” issue seems problematic in the framework considered here. This work also provides a complete solution to the CSL parametric oscillator system, a topic we suggest could play a very important role to further constrain the CSL parameters. Our results illustrate the remarkable power of inflation and cosmology to constrain new physics.

Figure 1

Wigner function of a squeezed quantum state at different times during inflation. Only the two-dimensional function corresponding to the set of variables $(v_{\mathit{k}}^{\mathrm{R}},p_{\mathit{k}}^{\mathrm{R}})$ has been represented, see Eq. (81). The time evolution of $\Re \mathrm{e}{\Omega }_{\mathit{k}}$ and $\Im \mathrm{m}{\Omega }_{\mathit{k}}$ has been expressed in terms of the two squeezing parameters $r_{\mathit{k}}$ and ${\varphi }_{\mathit{k}}$. These ones are given by the solutions (65, 66). The left upper panel corresponds to $r_{\mathrm{k}}=0.0005$ and the corresponding state is almost a coherent one. The right upper panel corresponds to $r_{\mathit{k}}=0.48$, the left bottom one to $r_{\mathit{k}}=0.88$ and, finally, the right bottom one to $r_{\mathit{k}}=2.31$. The effect of the squeezing and the cigar shape of Eq. (82) are clearly visible.

Figure 2

Wigner function of a coherent state (83), represented at different times during inflation. Contrary to the Wigner function of a squeezed state of Fig. 1, the shape remains unchanged during the cosmological evolution. The Wigner function just follows the classical trajectory, an ellipse here since we deal with an harmonic oscillator. This justifies the fact that a coherent state can be viewed as the “most classical quantum state.”

Figure 3

Spread in position and momentum for different values of $\gamma$ in the case of the harmonic oscillator; see Eq. (138). The conventional Schrödinger evolution corresponds to $\gamma =0$ and is represented by the black curve which oscillates with constant amplitude. On the contrary, when the collapse mechanism is turned on, the oscillations are damped (blue and red curves), the spreads tend toward a constant value and localization occurs.

Figure 4

Evolution of various physical length scales with time during the cosmic history in the CSL model described by Eq. (143) with a zoom on the transition from inflation to reheating inserted (see the concluding section). The solid line represents the Hubble radius ${\ell }_{\mathrm{H}}$ and the dashed-dotted green and red lines, the physical wavelengths of two Fourier modes of cosmological relevance today. The solid blue line represents the built-in CSL scale ${\ell }_{\gamma }$, see the discussion above Eq. (157). It is a preferred comoving scale and can also be viewed as a time-dependent preferred physical scale. Therefore, when a mode is below (above) ${\ell }_{\gamma }$ it remains so during the whole history of the Universe as is clear from the plot. This means that, contrary to the Hubble scale, there is no “${\ell }_{\gamma }$ crossing” during the cosmic evolution. As a consequence, one expects the power spectrum to acquire a broken power-law shape, with two different branches, an expectation confirmed by the calculations in Sec. 5d.

Figure 5

Ratio of the power spectrum given by Eq. (172) to the standard power spectrum given by Eq. (47) for different values of the parameter $\gamma /k_0^2$ (and for $\beta =-2.01$, a value leading to a standard power spectrum close to scale invariance). The number of e-folds between Hubble radius crossing and the end of inflation (for the modes of cosmological interest today) has been taken to $\Delta N_{*}=60$ and the initial conditions have been chosen to be the adiabatic vacuum.

Figure 6

Ratio of the power spectrum given by Eq. (a5) ($p=1$) to the standard power spectrum given by Eq. (47) for different values of the parameter $\gamma {\ell }_0/k_0$.

Figure 7

Ratio of the power spectrum given by Eq. (a9) ($p=2$) to the standard power spectrum given by Eq. (47) for different values of the parameter $\gamma {\ell }_0^2$.