Reconciliation of Dark Sector and Muon \((g-2)\) in the Light of Fast Expanding Universe

The lack of information before Big Bang Neucleosynthesis (BBN) allow us to assume the presence of a new species \(\phi \) whose energy density redshifts as \(a^{-(4+n)}\) where \(n>0\) and a is the scale factor. In this non-standard cosmological setup, we have considered \(U(1)_{L_\mu -L_\tau }\otimes U(1)_X\) gauge extension of the Standard Model (SM) and studied different phases of the cosmological evolution of a thermally decoupled dark sector such as leak-in, freeze-in, reannihilation, and late-time annihilation. This non-standard cosmological setup facilitates a larger portal coupling \((\epsilon )\) between the dark and the visible sectors even when the two sectors are not in thermal equilibrium. The dark sector couples with the \(\mu \) and \(\tau \) flavored leptons of the SM due to the tree level kinetic mixing between \(U(1)_X\) and \(U(1)_{L_\mu -L_\tau }\) gauge bosons. We show that in our scenario it is possible to reconcile the dark matter relic density and muon \((g-2)\) anomaly. In particular, we show that for \(3\times 10^{-4} \lesssim \epsilon \lesssim 10^{-3}\) , \(5.5{\,\mathrm MeV} \lesssim m_{Z^\prime } \lesssim 300{\,\mathrm MeV}\) , \(n=4\) , and \(1{\,\mathrm TeV} \lesssim m_\chi \lesssim 10{\,\mathrm TeV}\) relic density constraint of dark matter, constraint from muon \((g-2)\) anomaly, and other cosmological, astrophysical constraints are satisfied.