### 7.9 The cosmological constant problem

The condensed matter analogies offer us an important lesson concerning the cosmological constant
problem [660]. Sakharov’s induced gravity not only can give rise to an Einstein–Hilbert term under certain
conditions, it also gives rise to a cosmological term. This contribution would depend on the cut-off as ,
so if there were not additional contributions counterbalancing this term, emergent gravity in
condensed-matter-like systems will always give place to an enormous cosmological constant inducing a
strongly-repulsive force between quasiparticles.
However, we know that at low temperatures, depending on the microscopic characteristics of the system
we can have quite different situations. Remarkably, liquid systems (as opposed to gases) can remain stable
on their own, without requiring any external pressure. Their total internal pressure at equilibrium is
(modulo finite-size effects) always zero. This implies that, at zero temperature, if gravity emerges from a
liquid-like system, the total vacuum energy will be automatically forced to be
(relatively) small, and not a large number. The contribution coming from quasiparticle
fluctuations will be exactly balanced by contributions from the microphysics or “trans-Planckian”
contributions.

If the temperature is not zero there will be a pressure associated with the thermal
distribution of quasiparticles, which constitute the matter field of the system. Then, at equilibrium
one will have , so that there will be a small vacuum energy . This
value is not expected to match exactly the preferred value of obtained in the standard
cosmological model (CDM) as we are certain to be out of thermodynamic equilibrium. However, it
is remarkable that it matches its order of magnitude, (at least for the current epoch), albeit
any dynamical model implementing this idea will probably have to do so only at late times to
avoid possible tension with the observational data. Guided by these lessons, there are already a
number of heuristic investigations about how a cosmological term could dynamically adapt to the
evolution of the matter content, and which implications it could have for the evolution of the
universe [27, 350, 349, 351].

As final cautionary remark let us add that consideration of an explicit toy model for emergent
gravity [252] shows that the quantity that actually gravitates cannot be so easily predicted without an
explicit derivation of the analogue gravitational equations. In particular in [198] it was shown that the
relevant quantity entering the analogue of the cosmological constant is a contribution coming only from the
excitations above the condensate.