In quantum field theory the parameters of the vacuum action are subject to renormalization group running. In particular, the "cosmological constant" is not a constant in a quantum field theory context, still less should be zero. In this paper we continue with previous work, and derive the particle contributions to the running of the cosmological and gravitational constants in the framework of the Standard Model in curved space-time. At higher energies the calculation is performed in a sharp cut off approximation. We assess, in two different frameworks, whether the scaling dependences of the cosmological and gravitational constants spoil primordial nucleosynthesis. Finally, the cosmological implications of the running of the cosmological constant are discussed.

The cosmological constant (CC) term in Einstein's equations, Λ, was first associated to the idea of vacuum energy density. Notwithstanding, it is well-known that there is a huge, in fact appalling, discrepancy between the theoretical prediction and the observed value picked from the modern cosmological data. This is the famous, and extremely difficult, "CC problem". Paradoxically, the recent observation at the CERN Large Hadron Collider of a Higgs-like particle, should actually be considered ambivalent: on the one hand it appears as a likely great triumph of particle physics, but on the other hand it wide opens Pandora's box of the cosmological uproar, for it may provide (alas!) the experimental certification of the existence of the electroweak (EW) vacuum energy, and thus of the intriguing reality of the CC problem. Even if only counting on this contribution to the inventory of vacuum energies in the universe, the discrepancy with the cosmologically observed value is already of 55 orders of magnitude. This is the (hitherto) "real" magnitude of the CC problem, rather than the (too often) brandished 123 ones from the upper (but fully unexplored!) ultrahigh energy scales. Such is the baffling situation after 96 years of introducing the Λ-term by Einstein. In the following I will briefly (and hopefully pedagogically) fly over some of the old and new ideas on the CC problem. Since, however, the Higgs boson just knocked our door and recalled us that the vacuum energy may be a fully tangible concept in real phenomenology, I will exclusively address the CC problem from the original notion of vacuum energy, and its possible "running" with the expansion of the universe, rather than venturing into the numberless attempts to replace the CC by the multifarious concept of dark energy.

A large value of the cosmological constant (CC) is induced in the Standard Model (SM) of Elementary Particle Physics because of Spontaneous Symmetry Breaking. To provide a small value of the observable CC one has to introduce the vacuum term which cancels the induced one at some point in the very far infrared cosmic scale. Starting from this point we investigate whether the cancellation is preserved at different energy scales. We find that the running of the Higgs mass, couplings and the vacuum term inevitably result in a scaling dependence of the observable value. As a consequence one meets a nonzero CC at an energy scale comparable to the typical electron neutrino mass suggested by some experiments, and the order of magnitude of this constant is roughly the one derived from recent supernovae observations. However the sign of it is negative -opposite to what is suggested by these observations. This discrepancy may be a hint of the existence of an extra very light scalar, perhaps a Cosmon-like dilaton, which should essentially decouple from the SM Lagrangian, but that it nevertheless could mediate new macroscopic forces in the submillimeter range.

We investigate the properties of the FLRW flat cosmological models in which the vacuum energy density evolves with time, Λ(t). Using different versions of the Λ(t) model, namely quantum field vacuum, power series vacuum and power law vacuum, we find that the main cosmological functions such as the scale factor of the universe, the Hubble expansion rate H and the energy densities are defined analytically. Performing a joint likelihood analysis of the recent supernovae type Ia data, the Cosmic Microwave Background (CMB) shift parameter and the Baryonic Acoustic Oscillations (BAOs) traced by the Sloan Digital Sky Survey (SDSS) galaxies, we put tight constraints on the main cosmological parameters of the Λ(t) scenarios. Furthermore, we study the linear matter fluctuation field of the above vacuum models. We find that the patterns of the power series vacuum Λ = n1 H + n2 H 2 predict stronger small scale dynamics, which implies a faster growth rate of perturbations with respect to the other two vacuum cases (quantum field and power law), despite the fact that all the cosmological models share the same equation of state (EOS) parameter. In the case of the quantum field vacuum Λ = n0 + n2 H 2 , the corresponding matter fluctuation field resembles that of the traditional Λ cosmology. The power law vacuum (Λ ∝ a −n ) mimics the classical quintessence cosmology, the best fit being tilted in the phantom phase. In this framework, we compare the observed growth rate of clustering measured from the optical galaxies with those predicted by the current Λ(t) models. Performing a Kolmogorov-Smirnov (KS) statistical test we show that the cosmological models which contain a constant vacuum (ΛCDM), quantum field vacuum and power law vacuum provide growth rates that match well with the observed growth rate. However, this is not the case for the power series vacuum models (in particular, the frequently adduced Λ ∝ H model) in which clusters form at significantly earlier times (z ≥ 4) with respect to all other models (z ∼ 2). Finally, we derived the theoretically predicted dark-matter halo mass function and the corresponding distribution of cluster-size halos for all the models studied. Their expected redshift distribution indicates that it will be difficult to distinguish the closely resembling models (constant vacuum, quantum field and power-law vacuum), using realistic future X-ray surveys of cluster abundances. However, cluster surveys based on the Sunayev-Zeldovich detection method give some hope to distinguish the closely resembling models at high redshifts.PACS numbers: 98.80.-k, 95.35.+d, 95.36.+x

The renormalization group (RG) approach to cosmology is an efficient method to study the possible evolution of the cosmological parameters from the point of view of quantum field theory (QFT) in curved space-time. In this work we continue our previous investigations of the RG method based on potential low-energy effects induced from physics at very high energy scales M X M P . In the present instance we assume that both the Newton constant, G, and the cosmological term, Λ, can be functions of a scale parameter µ. It turns out that G(µ) evolves according to a logarithmic law which may lead to asymptotic freedom of gravity, similar to the gauge coupling in QCD. At the same time Λ(µ) evolves quadratically with µ. We study the consistency and cosmological consequences of these laws when µ ≃ H. Furthermore, we propose to extend this method to the astrophysical domain after identifying the local RG scale at the galactic level. It turns out that Kepler's third law of celestial mechanics receives quantum corrections that may help to explain the flat rotation curves of the galaxies without introducing the dark matter hypothesis. The origin of these effects (cosmological and astrophysical) could be linked, in our framework, to physics at M X ∼ 10 16−17 GeV .

We construct a semiclassical Friedmann-Lemaître-Robertson-Walker (FLRW) cosmological model assuming a running cosmological constant (CC). It turns out that the CC becomes variable at arbitrarily low energies due to the remnant quantum effects of the heaviest particles, e.g. the Planck scale physics. These effects are universal in the sense that they lead to a low-energy structure common to a large class of high-energy theories. Remarkably, the uncertainty concerning the unknown high-energy dynamics is accumulated into a single parameter ν, such that the model has an essential predictive power. Future Type Ia supernovae experiments (like SNAP) can verify whether this framework is correct. For the flat FLRW case and a moderate value ν ∼ 10 −2 , we predict an increase of 10 − 20% in the value of Ω Λ at redshifts z = 1 − 1.5 perfectly reachable by SNAP.

The discovery of dark energy (DE) as the physical cause for the accelerated expansion of the Universe is the most remarkable experimental finding of modern cosmology. However, it leads to insurmountable theoretical difficulties from the point of view of fundamental physics. Inflation, on the other hand, constitutes another crucial ingredient, which seems necessary to solve other cosmological conundrums and provides the primeval quantum seeds for structure formation. One may wonder if there is any deep relationship between these two paradigms. In this work, we suggest that the existence of the DE in the present Universe could be linked to the quantum field theoretical mechanism that may have triggered primordial inflation in the early Universe. This mechanism, based on quantum conformal symmetry, induces a logarithmic, asymptotically-free, running of the gravitational coupling. If this evolution persists in the present Universe, and if matter is conserved, the general covariance of Einstein's equations demands the existence of dynamical DE in the form of a running cosmological term, Λ, whose variation follows a power law of the redshift.

Despite the many outstanding cosmological observations leading to a strong evidence for a nonvanishing cosmological constant (CC) term Λ in the gravitational field equations, the theoretical status of this quantity seems to be lagging well behind the observational successes. It thus seems timely to revisit some fundamental aspects of the CC term in Quantum Field Theory (QFT). We emphasize that, in curved space-time, nothing a priori prevents this term from potentially having a mild running behavior associated to quantum effects. Remarkably, this could be the very origin of the dynamical nature of the Dark Energy, in contrast to many other popular options considered in the literature. In discussing this possibility, we also address some recent criticisms concerning the possibility of such running. Our conclusion is that, while there is no comprehensive proof of the CC running, there is no proof of the non-running either. The problem can be solved only through a deeper understanding of the vacuum contributions of massive quantum fields on a curved spacetime background. We suggest that such investigations are at the heart of one of the most important endeavors of fundamental theoretical cosmology in the years to come.