  Topological matter with Weyl points, such as superfluid 3He-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, where the "value of this constant cannot
be easily predicted by just looking at the ground state energy of the
microscopic system from which spacetime and its dynamics should emerge" (S.
Finazzi, S. Liberati and L. Sindoni, The cosmological constant: a lesson from
Bose-Einstein condensates, Phys. Rev. Lett. 108, 071101 (2012)). 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 3He-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 3He-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 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.
