The cosmological constant: a lesson from the effective gravity of topological Weyl media

Topological matter with Weyl points, such as superfluid $^3$He-A, provide an explicit example where there is a direct connection between the properly determined vacuum energy and the cosmological constant of the effective gravity emerging in condensed matter. This is in contrast to the acoustic gravity emerging in Bose–Einstein condensates (S. Finazzi, S. Liberati and L. Sindoni, Phys. Rev. Lett. 108, 071101 (2012); arXiv:1103.4841). The advantage of topological matter is that the relativistic fermions and gauge bosons emerging near the Weyl point obey the same effective metric and thus the effective gravity is more closely related to real gravity. We study this connection in the bi-metric gravity emerging in $^3$He-A, and its relation to the graviton masses, by comparison with a fully relativistic bi-metric theory of gravity. This shows that the parameter $\lambda$, which in $^3$He-A is the bi-metric generalization of the cosmological constant, coincides with the difference in the proper energy of the vacuum in two states (the nonequilibrium state without gravity and the equilibrium state in which gravity emerges) and is on the order of the characteristic Planck energy scale of the system. Although the cosmological constant $\lambda$ is huge, the cosmological term $T^{\Lambda}_{\mu\nu}$ itself is naturally non-constant and vanishes in the equilibrium vacuum, as dictated by thermodynamics. This suggests that the equilibrium state of any system including the final state of the Universe is not gravitating.