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

G. Jannes^{+}, G. E. Volovik ^{+*}

^{+}Low Temperature Laboratory, Aalto University, School of Science and Technology, P.O. Box 15100, FI-00076 AALTO, Finland

^{*}Landau Institute for Theoretical Physics of the RAS, 119334 Moscow, Russia

**Abstract**

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 λ, 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 λ is huge, the cosmological
term
T^{Λ}_{μν} 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.