Cosmic Microwave Background Constraints Cast a Shadow On Continuous Spontaneous Localization Models

The continuous spontaneous localization model solves the measurement problem of standard quantum mechanics by coupling the mass density of a quantum system to a white-noise field. Since the mass density is not uniquely defined in general relativity, this model is ambiguous when applied to cosmology. We however show that most natural choices of the density contrast already make current measurements of the cosmic microwave background incompatible with other laboratory experiments.

Figure 1

Time evolution of the physical distances at play in the early Universe. During inflation, the Hubble radius $H^{-1}$ (magenta line) is almost constant and, due to the expansion, the wavelength ${\lambda }_k$ of a Fourier mode (black line) for a given quantum field crosses out that scale, above which spacetime curvature sources parametric amplification. In the subsequent Universe, $H^{-1}$ increases faster than the scale factor $a$; hence, ${\lambda }_k$ crosses the Hubble radius back in. Depending on the value of $r_c$, ${\lambda }_k$ may cross out $r_c$ either during inflation ($r_c$) or during the radiation era ($r_c^{\prime }$).

Figure 2

Relevant values for $\gamma$. If $\gamma <{\gamma }_{\min }$, the wave function does not collapse and the power spectrum vanishes. If $\gamma >{\gamma }_{\max }$, the wave function collapses but the Born rule is violated and a non–scale-invariant power spectrum is obtained, which is excluded by the CMB observations. The region where ${\gamma }_{\min }<\gamma <{\gamma }_{\max }$, unbarred in Fig. 3, is where the wave function collapses to a scale-invariant power spectrum.

Figure 3

Observational constraints on the two parameters $r_c$ and $\lambda$ of the CSL model. The white region is allowed by laboratory experiments, while the unbarred region is allowed by CMB measurements (one uses $\mathrm{\Delta }N=50$ for the pivot scale of the CMB, $H_{\inf }={10}^{-5}{M}_{\mathrm{Pl}}$ and ${\epsilon }_1=0.005$). The two allowed regions are incompatible. The green dashed line stands for the upper bound on $\lambda$ if inflation proceeds at the Big-Bang nucleosynthesis (BBN) scale.