Running Vacuum Model Research Articles

Phantom Matter: A Challenging Solution to the Cosmological Tensions

The idea of composite dark energy (DE) is quite natural since on general grounds we expect that the vacuum energy density (associated with the cosmological term Λ) may appear in combination with other effective forms of DE, denoted X. Here we deal with model wXCDM, a simplified version of the old ΛXCDM model, and exploit the possibility that X behaves as “phantom matter” (PM), which appears in stringy versions of the running vacuum model (RVM). Unlike phantom DE, the PM fluid satisfies the strong energy condition like usual matter, hence bringing to bear positive pressure at the expense of negative energy. Bubbles of PM may appear in the manner of a transitory “phantom vacuum” tunneled into the late Universe before it heads toward a new de Sitter era, thereby offering a crop field for the growing of structures earlier than expected. Using Type Ia supernovae, cosmic chronometers, transversal baryon acoustic oscillations (BAO 2D), large-scale structure data, and the full cosmic microwave background likelihood from Planck 2018, we find that the H 0 and growth tensions virtually disappear, provided that BAO 2D are the only source of BAO data used in the fit. In contrast, our preliminary analysis using exclusively anisotropic BAO (BAO 3D) indicates that the ability to ease the H 0 tension is significantly reduced as compared to the scenario with BAO 2D, despite the fact that the overall fit to the cosmological data is still better than in the ΛCDM. Finally, our approach with BAO 2D favors quintessence-like behavior of the DE below z ≃ 1.5 at ≳3σ confidence level, which is compatible with the recent DESI measurements.

Discriminating interacting dark energy models using Statefinder diagnostic

In the present work, we perform a comparative study of different interacting dark energy (DE) models using the Statefinder diagnostics. In particular, 17 different forms of the energy transfer rate Q between DE and dark matter (DM) were focused on, belonging to the following categories: (i) linear models in energy densities of DE and DM, (ii) non-linear models, (iii) models with a change of direction of energy transfer between DE and DM, (iv) models involving derivatives of the energy densities, (v) parametrized interactions through a function of the coincidence parameter r~\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ilde{r}$$\\end{document}, and finally we also consider (vi) two kinds of models with a self-interaction between DM, without DE. These models have been already studied in the literature and constrained with observational data available at that time. In order to discriminate between them at background level, we use the Statefinder diagnostic, based on the computation and study of the so-called Statefinder parameters r, s in addition to the deceleration parameter q. We plot the evolution trajectories for the several interacting models on the r-q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r-q$$\\end{document}, r-s\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r-s$$\\end{document} planes, and we find some distinctive features and departures from Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document}CDM and other DE models, as Quintessence, Chaplygin Gas, running vacuum models (RVM) and Galileon.

Running vacuum in Brans & Dicke theory: A possible cure for the [formula omitted] and [formula omitted] tensions

Extensions of the gravitational framework of Brans & Dicke (BD) are studied by considering two different scenarios: (i) ‘BD-ΛCDM’, in which a rigid cosmological constant, Λ, is included, thus constituting a BD version of the vanilla concordance ΛCDM model (the current standard model of cosmology with flat three-dimensional geometry), and (ii) ‘BD-RVM’, a generalization of (i) in which the vacuum energy density (VED), ρvac, is a running quantity evolving with the square of the Hubble rate: δρvac(H)∝νmPl2(H2−H02) (with |ν|≪1 and H0 being the present value of the Hubble rate). This dynamical scenario is motivated by recent studies of quantum field theory (QFT) in curved spacetime, which lead to the running vacuum model (RVM). In both cases, rigid or running ρvac, the GR limit can be recovered smoothly. We solve the background as well as the perturbation equations for each cosmological model and test their performance against the modern wealth of cosmological data, namely a compilation of the latest SNIa+H(z)+BAO+LSS+CMB observations. We utilize the AIC and DIC statistical information criteria in order to determine if they can fit better the observations than the concordance model. The two BD extensions are tested by considering three different datasets. According to the AIC and DIC criteria, both BD extensions (i) and (ii) are competitive, but the second one (the BD-RVM scenario) is particularly favored when it is compared with the vanilla model. This fact may indicate that the current observations favor a mild dynamical evolution of the Newtonian coupling GN as well as of the VED. While further studies will be necessary, the results presented here suggest that the Brans & Dicke theory with running vacuum could have the potential to alleviate the two tensions at the same time.

Stringy running vacuum model and current tensions in cosmology

We discuss the potential alleviation of both the Hubble and the growth of galactic structure data tensions observed in the current epoch of cosmology in the context of the so-called stringy running vacuum model (RVM) of cosmology. This is a gravitational field theory coupled to matter, which, at early eras, contains gravitational (Chern–Simons (CS) type) anomalies and torsion, arising from the fundamental degrees of freedom of the massless gravitational multiplet of an underlying microscopic string theory. The model leads to RVM type inflation without external inflatons, arising from the quartic powers of the Hubble parameter that characterize the vacuum energy density due to primordial-gravitational-wave-induced anomaly CS condensates, and dominate the inflationary era. In modern eras, of relevance to this work, the gravitational anomalies are cancelled by chiral matter, generated at the end of the RVM inflationary era, but cosmic radiation and other matter fields are still responsible for a RVM energy density with terms exhibiting a quadratic-power-of-Hubble-parameter dependence, but also products of the latter with logarithmic H-dependencies, arising from potential quantum-gravity and quantum-matter loop effects. In this work, such terms are examined phenomenologically from the point of view of the potential alleviation of the aforementioned current tensions in cosmology. Using standard information criteria, we find that these tensions can be substantially alleviated in a way consistent not only with the data, but also with the underlying microscopic theory predictions, associated with the primordial dynamical breaking of supergravity that characterize a pre-RVM-inflationary phase of the model.

Exploring models of running vacuum energy with viscous dark matter from a dynamical system perspective

Running vacuum models and viscous dark matter scenarios beyond perfect fluid idealization are two appealing theoretical strategies that have been separately studied as alternatives to solve some problems rooted in the ΛCDM cosmological model. In this paper, we combine these two notions in a single cosmological setting and investigate their cosmological implications, paying particular attention in the interplay between these two constituents in different cosmological periods. Specifically, we consider a well-studied running vacuum model inspired by renormalization group, and a recently proposed general parameterization for the bulk viscosity ξ. By employing dynamical system analysis, we explore the physical aspects of the new phase space that emerges from the combined models and derive stability conditions that ensure complete cosmological dynamics. We identify four distinct classes of models and find that the critical points of the phase space are non-trivially renewed compared to the single scenarios. We then proceed, in a joint and complementary way to the dynamical system analysis, with a detailed numerical exploration to quantify the impact of both the running parameter and the bulk viscosity coefficient on the cosmological evolution. Thus, for some values of the model parameters, numerical solutions show qualitative differences from the ΛCDM model, which is phenomenologically appealing in light of cosmological observations.

Running vacuum in QFT in FLRW spacetime: the dynamics of rho _{textrm{vac}}(H) from the quantized matter fields

Phenomenological work in the last few years has provided significant support to the idea that the vacuum energy density (VED) is a running quantity with the cosmological evolution and that this running helps to alleviate the cosmological tensions afflicting the Lambda CDM. On the theoretical side, recent devoted studies have shown that the properly renormalized rho _{textrm{vac}} in QFT in FLRW spacetime adopts the ‘running vacuum model’ (RVM) form. While in three previous studies by two of us (CMP and JSP) such computations focused solely on scalar fields non-minimally coupled to gravity, in the present work we compute the spin-1/2 fermionic contributions and combine them both. The calculation is performed using a new version of the adiabatic renormalization procedure based on subtracting the UV divergences at an off-shell renormalization point M. The quantum scaling of rho _{textrm{vac}} with M turns into cosmic evolution with the Hubble rate, H. As a result the ‘cosmological constant’ Lambda appears in our framework as the nearly sustained value of 8pi G(H)rho _{textrm{vac}}(H) around (any) given epoch H, where G(H) is the gravitational coupling, which is also running, although very mildly (logarithmically). We find that the VED evolution at present reads delta rho _textrm{vac}(H)sim nu _{textrm{eff}}, m_{textrm{Pl}}^2 left( H^2-H_0^2 right) (|nu _{textrm{eff}}|ll 1). The coefficient nu _{textrm{eff}} receives contributions from all the quantized fields, bosons and fermions, which we compute here for an arbitrary number of matter fields. Remarkably, there also exist higher powers mathcal{O}(H^{6}) which can trigger inflation in the early universe. Finally, the equation of state (EoS) of the vacuum receives also quantum corrections from bosons and fermion fields, shifting its value from − 1. The striking consequence is that the EoS of the quantum vacuum may nowadays effectively appears as quintessence.

Anomalies, the Dark Universe and Matter-Antimatter asymmetry

I review a (3+1)-dimensional, string-inspired cosmological model with gravitational anomalies (of Chern-Simons (CS) type) at early epochs, and a totally-antisymmetric torsion, dual to a massless axion-like field (“gravitational axion”), which couples to the CS term. Under appropriate conditions, primordial gravitational waves can condense, leading to a condensate of the CS anomaly term. As a consequence, one obtains inflation in this theory, of running-vacuum-model (RVM) type, without the need for external inflatons. At the end of the inflationary era, chiral fermionic matter is generated, whose gravitational anomalies cancel the primordial ones. On the other hand, chiral anomalies of gauge type, which are also generated by the chiral matter, remain present during the post-inflationary epochs and become responsible for the generation of a non-perturbative mass for the torsion-related gravitational axion, which, in this way, might play the rôle of a Dark Matter component of geometrical origin. Moreover, in this model, stringy non-perturbative effects during the RVM inflationary phase generate periodic structures for the potential of axion-like particles that arise due to compactification, and co-exist with the gravitational axions. Such periodic potential modulations may lead to an enhanced production of primordial black holes during inflation, which in turn affects the profile of the generated gravitational waves during the radiation era, with potentially observable consequences. This model also entails an unconventional mechanism for Leptogenesis, due to Lorentz-violating backgrounds of the gravitational axions that are generated during inflation, as a consequence of the anomaly condensates, and remain undiluted in the radiation era.

Running Vacuum in the Universe: Phenomenological Status in Light of the Latest Observations, and Its Impact on the σ8 and H0 Tensions

A substantial body of phenomenological and theoretical work over the last few years strengthens the possibility that the vacuum energy density (VED) of the universe is dynamical, and in particular that it adopts the ‘running vacuum model’ (RVM) form, in which the VED evolves mildly as δρvac(H)∼νeffmPl2OH2, where H is the Hubble rate and νeff is a (small) free parameter. This dynamical scenario is grounded on recent studies of quantum field theory (QFT) in curved spacetime and also on string theory. It turns out that what we call the ‘cosmological constant’, Λ, is no longer a rigid parameter but the nearly sustained value of 8πG(H)ρvac(H) around any given epoch H(t), where G(H) is the gravitational coupling, which can also be very mildly running (logarithmically). Of particular interest is the possibility suggested in past works that such a running may help to cure the cosmological tensions afflicting the ΛCDM. In the current study, we reanalyze the RVM in full and we find it becomes further buttressed. Using modern cosmological data, namely a compilation of the latest SNIa+BAO+H(z)+LSS+CMB observations, we probe to what extent the RVM provides a quality fit better than the concordance ΛCDM model, with particular emphasis on its impact on the σ8 and H0 tensions. We utilize the Einstein–Boltzmann system solver CLASS and the Monte Carlo sampler MontePython for the statistical analysis, as well as the statistical DIC criterion to compare the running vacuum against the rigid vacuum (νeff=0). On fundamental grounds, νeff receives contributions from all the quantized matter fields in FLRW spacetime. We show that with a tiny amount of vacuum dynamics (νeff≪1) the global fit can improve significantly with respect to the ΛCDM and the mentioned tensions may subside to inconspicuous levels.

Running vacuum versus holographic dark energy: a cosmographic comparison

We perform a comparative study of different types of dynamical dark energy models (DDEs) using the cosmographic method. Among the models being examined herein we have the Running Vacuum models (RVMs), which have demonstrated considerable ability to fit the overall cosmological data at a level comparable to the standard cosmological model, Lambda CDM, and capable of alleviating the sigma _8 and H_0 tensions. At the same time we address a variety of Holographic dark energy models (HDEs) with different options for the time (redshift)-varying model parameter c=c(z). We deal with the HDEs under the double assumption of fixed and evolving holographic length scale and assess which one is better. Both types of DDEs (RVMs and HDEs) are confronted with the most robust cosmographic data available, namely the Pantheon sample of supernovae data (SnIa), the baryonic acoustic oscillations data (BAOs) extracted from measurement of the power spectrum and bispectrum of the BOSS data release, and the cosmic chronometer measurements of the Hubble rate (CCHs) at different redshifts obtained from spectroscopic observations of passively evolving galaxies. Using these data samples we assess the viability of the mentioned DDEs and compare them with the concordance Lambda CDM model. From our cosmographic analysis we conclude that the RVMs fare comparably well to the Lambda CDM, a fact which adds up more credit to their sound phenomenological status. In contrast, while some of the HDEs are favored using the current Hubble horizon as fixed holographic length, they become highly unfavoured in the more realistic case when the holographic length is dynamical and evolves as the Hubble horizon.

Gravitational waves with dark energy

In this article, we study the tensor mode equation of perturbation in the presence of nonzero−Λ as dark energy, the dynamic nature of which depends on the Hubble parameter H and/or its time derivative. Dark energy, according to the total vacuum contribution, has a slight effect during the radiation-dominated era, but it reduces the squared amplitude of gravitational waves (GWs) up to 60% for the wavelengths that enter the horizon during the matter-dominated era. Moreover, the observations bound on dark energy models, such as running vacuum model (RVM), generalized running vacuum model (GRVM) and generalized running vacuum subcase (GRVS), are effective in reducing the GWs' amplitude. Although this effect is less for the wavelengths that enter the horizon at later times, this reduction is stable and permanent.

The cosmological constant problem and running vacuum in the expanding universe.

It is well known that quantum field theory (QFT) induces a huge value of the cosmological constant (CC), [Formula: see text], which is outrageously inconsistent with cosmological observations. We review here some aspects of this fundamental theoretical conundrum (the cosmological constant problem, CCP) and strongly argue in favour of the possibility that the cosmic vacuum density [Formula: see text] may be mildly evolving with the expansion rate [Formula: see text]. Such a 'running vacuum model' (RVM) proposal predicts an effective dynamical dark energy without postulating new ad hoc fields (quintessence and the like). Using the method of adiabatic renormalization within QFT in curved space-time, we find that [Formula: see text] acquires a dynamical component [Formula: see text] caused by the quantum matter effects. There are also [Formula: see text] ([Formula: see text]) contributions, some of which may trigger inflation in the early universe. Remarkably, the evolution of the adiabatically renormalized [Formula: see text] is not affected by dangerous terms proportional to the quartic power of the masses ([Formula: see text]) of the quantized matter fields. Traditionally, these terms have been the main source of trouble as they are responsible for the extreme fine-tuning feature of the CCP. In the context under study, however, the late time [Formula: see text] around [Formula: see text] is given by a dominant term ([Formula: see text]) plus the aforementioned mild dynamical component [Formula: see text] (with [Formula: see text]), which makes the RVM to mimic quintessence. Finally, on the phenomenological side, we show that the RVM may be instrumental in alleviating some of the most challenging problems (so-called 'tensions') afflicting nowadays the observational consistency of the 'concordance' [Formula: see text]CDM model, such as the [Formula: see text] and [Formula: see text] tensions. This article is part of the theme issue 'The future of mathematical cosmology, Volume 2'.

Unifying turnaround-bounce cosmology in a cyclic universe considering a running vacuum model

Among many models which can describe the bouncing cosmology, a matter bounce scenario that is deformed by a running vacuum model of dark energy (RVM-DE) has been interested. In this research, I show that a class of RVM-CDM (cold dark matter) model can also describe a cyclical cosmology in which the universe undergoes cycles of expansion to the contraction phase and vice versa. To this end, following our previous work, I consider one of the most successful classes of RVM-CDM model in bouncing cosmology, [Formula: see text], in which the power spectral index gets a red tilt and the running of the spectral index may give a negative value by choosing the appropriate value of parameters [Formula: see text], [Formula: see text], [Formula: see text], which is consistent with the cosmological observations. It is worthwhile to mention that most matter bounce models do not produce this negative value. However, the main purpose of this paper is to investigate the RVM-CDM model in the turnaround phase. Far from the bounce in a phantom expanding universe, the turnaround conditions are investigated before the occurrence of a sudden big rip. By analyzing the Hubble parameter, equation of state (EoS) parameter, and deceleration parameter around the turnaround, we show that a successful turnaround may occur after an expansion in an interacting case of RVM-CDM by choosing the appropriate value of parameters. A minimum value for the interaction parameter is obtained and also find any relation between other model parameters. Finally, the effect of each parameter on a turnaround is studied, and we see that the transition time from accelerating to decelerating expansion can occur earlier for larger values of interaction parameter. Also, in several graphs, the effect of the second term in DE density, including [Formula: see text], is studied, and we see that by increasing its coefficient, [Formula: see text], the transition point leads to lower values.

Renormalizing the vacuum energy in cosmological spacetime: implications for the cosmological constant problem

The renormalization of the vacuum energy in quantum field theory (QFT) is usually plagued with theoretical conundrums related not only with the renormalization procedure itself, but also with the fact that the final result leads usually to very large (finite) contributions incompatible with the measured value of Lambda in cosmology. As a consequence, one is bound to extreme fine-tuning of the parameters and so to sheer unnaturalness of the result and of the entire approach. We may however get over this adversity using a different perspective. Herein, we compute the zero-point energy (ZPE) for a nonminimally coupled (massive) scalar field in FLRW spacetime using the off-shell adiabatic renormalization technique employed in previous work. The on-shell renormalized result first appears at sixth adiabatic order, so the calculation is rather cumbersome. The general off-shell result yields a smooth function rho _{mathrm{vac}}(H) made out of powers of the Hubble rate and/or of its time derivatives involving different (even) adiabatic orders sim H^N (N=0, 2,4,6,ldots ), i.e. it leads, remarkably enough, to the running vacuum model (RVM) structure. We have verified the same result from the effective action formalism and used it to find the beta -function of the running quantum vacuum. No undesired contributions sim m^4 from particle masses appear and hence no fine-tuning of the parameters is needed in rho _{mathrm{vac}}(H). Furthermore, we find that the higher power sim H^6 could naturally drive RVM-inflation in the early universe. Our calculation also elucidates in detail the equation of state of the quantum vacuum: it proves to be not exactly -1 and is moderately dynamical. The form of rho _{mathrm{vac}}(H) at low energies is also characteristic of the RVM and consists of an additive term (the so-called ‘cosmological constant’) together with a small dynamical component sim nu H^2 (|nu |ll 1). Finally, we predict a slow (sim ln H) running of Newton’s gravitational coupling G(H). The physical outcome of our semiclassical QFT calculation is revealing: today’s cosmic vacuum and the gravitational strength should be both mildly dynamical.

Power-law holographic dark energy and cosmology

We formulate power-law holographic dark energy, which is a modified holographic dark energy model based on the extended entropy relation arising from the consideration of state mixing between the ground and the excited ones in the calculation of the entanglement entropy. We construct two cases of the scenario, imposing the usual future event horizon choice, as well as the Hubble one. Thus, the former model is a one-parameter extension of standard holographic dark energy, recovering it in the limit where power-law extended entropy recovers Bekenstein–Hawking one, while the latter belongs to the class of running vacuum models, a feature that may reveal the connection between holography and the renormalization group running. For both models we extract the differential equation that determines the evolution of the dark-energy density parameter and we provide the expression for the corresponding equation-of-state parameter. We find that the scenario can describe the sequence of epochs in the Universe evolution, namely the domination of matter followed by the domination of dark energy. Moreover, the dark-energy equation of state presents a rich behavior, lying in the quintessence regime or passing into the phantom one too, depending on the values of the two model parameters, a behavior that is richer than the one of standard holographic dark energy.

Big bang nucleosynthesis constraints on higher-order modified gravities

We use Big Bang Nucleosynthesis (BBN) data in order to impose constraints on higher-order modified gravity, and in particular on: (i) $f(G)$ Gauss-Bonnet gravity, and $f(P)$ cubic gravities, arising respectively through the use of the quadratic-curvature Gauss-Bonnet $G$ term, and the cubic-curvature combination, (ii) string-inspired quadratic Gauss-Bonnet gravity coupled to the dilaton field, (iii) models with string-inspired quartic curvature corrections, and (iv) running vacuum models. We perform a detailed investigation of the BBN epoch and we calculate the deviations of the freeze-out temperature $T_f$ in comparison to $\Lambda$CDM paradigm. We then use the observational bound on $ \left|\frac{\delta {T}_f}{{T}_f}\right|$ in order to extract constraints on the involved parameters of various models. We find that all models can satisfy the BBN constraints and thus they constitute viable cosmological scenarios, since they can additionally account for the dark energy sector and the late-time acceleration, in a quantitative manner, without spoiling the formation of light elements during the BBN epoch. Nevertheless, the obtained constraints on the relevant model parameters are quite strong.

Constraints on running vacuum models with the baryon-to-photon ratio

We study the influence of running vacuum on the baryon-to-photon ratio in running vacuum models (RVMs). The photon and baryon number densities, from photon decoupling to the present day, are obtained in the context of RVMs by assuming that photons and baryons can be coupled to running vacuum, respectively. Both cases lead to a time-evolving baryon-to-photon ratio, whose values at different epochs of the universe are strictly constrained by observations. It is found that if the dominant dynamical term of running vacuum is indeed coupled to photons or baryons, the corresponding coefficient must be far less than the upper limit given by previous research, and so it is difficult to distinguish RVMs from the standard {varLambda }CDM model. Such RVMs which are barely distinguishable from the {varLambda }CDM model seem to be unconvincing. Therefore, if RVMs are reliable cosmological models, running vacuum is unlikely to be coupled to photons or baryons, and then the coefficient of the dominant dynamical term of running vacuum no longer needs to be extremely small.

A Real Scalar Field Unifying the Early Inflation and the Late Accelerating Expansion of the Universe through a Quadratic Equation of State: The Vacuumon

In a previous paper we introduced a cosmological model describing the early inflation, the intermediate decelerated expansion, and the late accelerating expansion of the universe in terms of a single barotropic fluid characterized by a quadratic equation of state. We obtained a scalar field representation of this fluid and determined the potential V(ϕ) connecting the inflaton potential in the early universe to the quintessence potential in the late universe. This scalar field has later been called the ‘vacuumon’ by other authors, in the context of the Running Vacuum model. In this paper, we study how the scalar field potential is modified by the presence of other cosmic components such as stiff matter, black-body radiation, baryonic matter, and dark matter. We also determine the mass m and the self-interaction constant λ of the scalar field given by the second and fourth derivatives of the potential at its extrema. We find that its mass is imaginary in the early universe with a modulus of the order of the Planck mass MP=(ℏc/G)1/2=1.22×1019GeV/c2 and real in the late universe with a value of the order of the cosmon mass mΛ=(Λℏ2/c4)1/2=2.08×10−33eV/c2 predicted by string theory. Although our model is able to describe the evolution of the homogeneous background for all times, it cannot account for the spectrum of fluctuations in the early universe. Indeed, by applying the Hamilton–Jacobi formalism to our model of early inflation, we find that the Hubble hierarchy parameters and the spectral indices lead to severe discrepancies with the observations. This suggests that the vacuumon potential is just an effective classical potential that cannot be directly used to compute the fluctuations in the early universe. A fully quantum field theory may be required to achieve that goal. Finally, we discuss the connection between our model based on a quadratic equation of state and the Running Vacuum model which assumes a variation of the cosmological constant with the Hubble parameter.

Running vacuum model versus ΛCDM – a Bayesian analysis

ABSTRACT We study the significance of the running vacuum model in which the vacuum energy density depends on the square of Hubble parameter, in comparison with the ΛCDM model. The Bayesian inference method is employed to appraise the relative significance of the running vacuum model, using the combined data sets, SN1a+CMB+BAO and SN1a+CMB+BAO+OHD. The model parameters and the corresponding errors are estimated from the marginal probability density function of the model parameters. The parameter that distinguish the running vacuum model from the ΛCDM model is ν. With the SN1a+CMB+BAO data set, we have found that the parameter ν is different from zero at ∼2.7σ. With the second data set, SN1a+CMB+BAO+OHD, the significance improved considerably to 3.4σ. Marginalizing over all model parameters with suitable prior, we have obtained the Bayes factor as the ratio of Bayesian evidence of our model and the ΛCDM model. The analysis based on Jeffrey’s scale of bayesian inference shows that the evidence of our model against the ΛCDM model is weak for the data set SN1a+CMB+BAO. We have obtained a definite evidence of running vacuum model for SN1a+CMB+BAO + OHD data set. This indicates that the dark energy could be dynamical.

Torsion in String-Inspired Cosmologies and the Universe Dark Sector

Several aspects of torsion in string-inspired cosmologies are reviewed. In particular, its connection with fundamental, string-model independent, axion fields associated with the massless gravitational multiplet of the string are discussed. It is argued in favour of the role of primordial gravitational anomalies coupled to such axions in inducing inflation of a type encountered in the “Running-Vacuum-Model (RVM)” cosmological framework, without fundamental inflaton fields. The gravitational-anomaly terms owe their existence to the Green–Schwarz mechanism for the (extra-dimensional) anomaly cancellation, and may be non-trivial in such theories in the presence of (primordial) gravitational waves at early stages of the four-dimensional string universe (after compactification). The paper also discusses how the torsion-induced stringy axions can acquire a mass in the post inflationary era, due to non-perturbative effects, thus having the potential to play the role of (a component of) dark matter in such models. Finally, the current-era phenomenology of this model is briefly described with emphasis placed on the possibility of alleviating tensions observed in the current-era cosmological data. A brief phenomenological comparison with other cosmological models in contorted geometries is also made.

Dynamical analysis of interacting running vacuum models in DGP braneworld

The time dependent vacuum energy provides new mechanism for describing the entire cosmic history (from inflation to dark energy) of the universe. In the framework of DGP braneworld theory, we discuss the dynamical properties of varying vacuum cosmology by considering the interaction between the cosmological fluids. We engage various functional forms (linear and non-linear) of interaction term and apply dynamical system approach. In each case, we compute critical points as well as examine their existence and stability. These cosmological solutions describe the different cosmological eras. It is observed that in each case, system admits the stable critical points for small values of interaction parameter b (strength of interaction) which is a favorable condition from the observational point of view. We also develop the phase space portraits of the interaction models and find attractive behavior of the universe for the majority cases of interaction models.