Relaxion measure problem

We examine the necessity of requiring that relaxion dynamics is dominated by classical slow roll and not quantum fluctuations. It has been recently proposed by Nelson and Prescod-Weinstein [Phys. Rev. D 96, 113007 (2017)] that abandoning this requirement can lead to a unified solution of the hierarchy and strong $CP$ problems in QCD relaxion models. In more general models this results in a higher value of the allowed cutoff. In this work we find, however, that relaxing this condition can result in the Universe being dominated in physical volume by regions arising from large quantum fluctuations of the relaxion. These regions turn out to be problematic for the relaxion mechanism because either the relaxion does not stabilize at all or it stabilizes at vacua which cannot reproduce the observed properties of our Universe. The size of these undesirable regions is moreover ambiguous because of the measure problem. For instance, we show that if one chooses to use the scale-factor cutoff measure such dangerous regions occupy a negligible volume and these issues do not arise.

Figure 1

Here is a schematic representation of the relaxion potential in Eq. (6) after setting $⟨H{⟩}^2=-{\mu }^2(\varphi )/\lambda$. The region marked $\mathrm{\Sigma }$ shows the vacua where the relaxion stabilizes during inflation taking into account its quantum spreading in Eq. (11). Even after its expectation value stabilizes, there can be regions where the relaxion has undergone a large quantum fluctuation to a generic point in field space such as ${\varphi }_i$ above. While such fluctuations are rare during inflation, the fact that the vacuum energy at ${\varphi }_i$ is much larger compared to that in the $\mathrm{\Sigma }$ region can amplify these fluctuations in volume after the inflaton settles down. In the NP model the size of the backreaction changes after inflation and this figure shows the backreaction potential before this happens. In this model, ${\varphi }_j$ is the position of the minima that have an $\mathcal{O}(1)$ relaxion stopping phase after the inflaton stabilizes and the barriers become large.