PeVatron Candidate SNR G106.3+2.7 in a Low-density Cavity: a Multiwavelength Test

In this paper, we constrain the density of the interstellar medium (ISM) around the hadronic PeVatron candidate, supernova remnant (SNR) G106.3+2.7, based on X-ray and $\gamma$-ray observations. The purpose of this investigation is to understand the influence of the gaseous environment on this SNR as a proton PeVatron candidate. By modelling the self-regulated propagation of the CRs injected from the SNR, we calculate the $\gamma$-ray emission of CRs via the hadronuclear interactions with the molecular cloud and the ISM, and use the measured $\gamma$-ray flux to constrain the ISM density around the SNR. Our results support the picture that the SNR is expanding into a low-density ($n<0.05 cm^{-3}$) cavity, enabling the SNR to be a potential proton PeVatron despite that it presently is not in the very early phase.