In quantum field theory the vacuum fluctuations of quantum fields are computed using regularisation and renormalisation techniques. The energy and pressure of the quantum vacuum cannot exactly be predicted, but instead, only their order of magnitude is suggested by the scale at which new physics set in, M.
In a cosmological setting the vacuum energy is dominated by the constant contribution ~M4 that also arises in flat space (Minkowski). Subdominant corrections of the order M2H2 arise in cosmology, where H(t) is the Hubble parameter that describes the expansion rate of the geometry. Usually, the leading term ~M4 is thought to be gravitating and therefore leads to the naturalness problem of the cosmological constant: the value suggested by quantum field theory is about 120 orders of magnitude larger than the value of the cosmological constant required to explain the observed accelerated expansion of the recent Universe.
We first clarify some recurrent confusion about regularising and renormalising the quantum vacuum with a sharp momentum cut-off. This leads to a better understanding of the role of the divergences: the leading M4 term is absorbed into the cosmological constant, while the subdominant M2H2 term can only be absorbed into Newton's constant if the energy-momentum tensor of zero-point fluctuations is considered to be conserved independently from all other contributions.
Then we discuss a recent proposal to avoid the naturalness problem, motivated by successful subtraction schemes like the Casimir effect and the ADM mass in asymptotically flat space-times. Maggiore (2011) proposed to define the cosmologically gravitating vacuum energy as the expectation value evaluated in the the FRW metric minus the vacuum energy in Minkowski space. Effectively, this means that the vacuum in Minkowski space does not gravitate and that the vacuum energy in FRW is ~M2H2. As a consequence, the naturalness problem is avoided and an avenue to interesting cosmological consequences of the vacuum energy is opened, if quantum fluctuations are not conserved on their own.
Finally, we provide an example where the quantum fluctutations are not conserved separately and therefore the vacuum energy cannot be absorbed into a renormalisation of Newton's constant. If an ultra-light scalar field is present with mass m<H(t), then it cannot be integrated out completely, in the language of effective field theory. We argue that integrating out only the heavy modes of the field yields an effective theory with a scalar representing the light modes that acquires a non-minimal coupling to gravity. We assess the phenomenology of such models and find that it generically leads to an early dark energy scenario, see e.g. Hollenstein et al. (2009).