In cosmology, phenomenologically motivated expressions for running vacuum are commonly parametrized as linear functions Λ(H 2 ) or Λ(R). Such kind of models assume an equation of state for vacuum given by P Λ = − ρ Λ , relating their background pressure P Λ and mean energy density ρ Λ ≡ Λ/8πG. This equation of state requires that the dynamic for vacuum is due to the energy exchange with the material species. Most of the approaches to background level consider only the energy exchange between vacuum and the transient dominant material component of the universe. We extend such models assuming the running vacuum as the sum of independent contributions ρ Λ = i ρ Λi , associated with (and interacting with) each of the i material species. We derive the linear scalar perturbations for two running scenarios, modeling its cosmic evolution and identifying their different imprints on the cosmic microwave background anisotropies and the matter power spectrum. In the Λ(H 2 ) scenario the running vacuum are coupled with all the material species in the universe, whereas the Λ(R) description only leads to coupling between vacuum and the non-relativistic matter components; which produces different imprints of the two models on the matter power spectrum. A comparison with the Planck 2015 data was made in order to constrain the free parameters of the models. In the case of the Λ(H 2 ) model, it was found that Ω Λ = 0.705 ± 0.027 and H 0 = 69.6 ± 2.9 km M pc −1 s −1 , which diminish the tension with the low redshift expectations. * elduartep@usp.br †

The dark-energy component of the Universe still remains a mystery; however, several papers based on observational data have shown that its equation of state may have an oscillatory behaviour. In this paper, we provide a general description for the dark-energy equation of state w(z) in the form of a Fourier series. This description generalizes some previous dynamical dark-energy models and is in agreement with the w(z) reconstructions. We make use of a modified version of a simple and fast Markov chain Monte Carlo code to constrain the model parameters. For the analysis we use data from supernovae type Ia, baryon acoustic oscillations, H(z) measurements and cosmic microwave background. We provide a comparison of the proposed model with ΛCDM, wCDM and the standard Taylor approximation. The Fourier-series expansion of w(z) is preferred from ΛCDM at more than the 3σ significance level based on the improvement in the fit alone. We use the Akaike criterion to perform the model comparison and find that, even though there are extra parameters, there is a slight preference for the Fourier series compared with the ΛCDM model. The preferred shape of w(z) found here puts in jeopardy the single scalar field models, as they cannot reproduce the crossing of the phantom divide line w = 1.

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We investigate the cosmological production of gravitational waves in a nonsingular flat cosmology powered by a "running vacuum" energy density described by ρ Λ ≡ ρ Λ (H), a phenomenological expression potentially linked with the renormalization group approach in quantum field theory in curved spacetimes. The model can be interpreted as a particular case of the class recently discussed by Perico et al. (Phys. Rev. D 88, 063531, 2013) which is termed complete in the sense that the cosmic evolution occurs between two extreme de Sitter stages (early and late time de Sitter phases). The gravitational wave equation is derived and its time-dependent part numerically integrated since the primordial de Sitter stage. The generated spectrum of gravitons is also compared with the standard calculations where an abrupt transition, from the early de Sitter to the radiation phase, is usually assumed. It is found that the stochastic background of gravitons is very similar to the one predicted by the cosmic concordance model plus inflation except at higher frequencies (ν 100 kHz). This remarkable signature of a "running vacuum" cosmology combined with the proposed high frequency gravitational wave detectors and measurements of the CMB polarization (Bmodes) may provide a new window to confront more conventional models of inflation.

The problem of cosmological production of gravitational waves is discussed in the framework of an expanding, spatially homogeneous and isotropic FRW type Universe with time-evolving vacuum energy density. The gravitational wave equation is established and its modified time-dependent part is analytically resolved for different epochs in the case of a flat geometry. Unlike the standard ΛCDM cosmology (no interacting vacuum), we show that gravitational waves are produced in the radiation era even in the context of general relativity. We also show that for all values of the free parameter, the high frequency modes are damped out even faster than in the standard cosmology both in the radiation and matter-vacuum dominated epoch. The formation of the stochastic background of gravitons and the remnant power spectrum generated at different cosmological eras are also explicitly evaluated. It is argued that measurements of the CMB polarization (B-modes) and its comparison with the rigid ΛCDM model plus the inflationary paradigm may become a crucial test for dynamical dark energy models in the near future.PACS number: 04.30.Db, 95.36.+x,

BackgroundThe economic cost of not breastfeeding in Indonesia is estimated at US$1.5–9.4 billion annually, the highest in South East Asia. Half of the 33.6 million working women of reproductive age (WRA) in Indonesia (15-49 years) are informal employees, meaning they are working as casual workers or they are self-employed (small scale business) and assisted by unpaid/family worker(s). No specific maternity protection entitlements are currently available for WRA working informally in Indonesia. This study aims to estimate the financing need of providing maternity leave cash transfer (MCT) for WRA working in the informal sector in Indonesia.MethodThe costing methodology used is the adapted version of the World Bank methodology by Vilar-Compte et al, following pre-set steps to estimate costs using national secondary data. We used the 2018 Indonesian National Socio-Economic Survey to estimate the number of women working informally who gave birth within the last year. The population covered, potential cash transfer’s unitary cost, the incremental coverage of the policy in terms of time and coverage, and the administrative costs were used to estimate the cost of MCT for the informal sector. ResultAt 100% coverage for 13 weeks of leave, the yearly financing need of MCT ranged from US$175million (US$152/woman) to US$669million (US$583/woman). The share of the yearly financing need did not exceed 0.5% of Indonesian Gross Domestic Product (GDP).ConclusionThe yearly financing need of providing MCT for eligible WRA working in the informal sector is economically attractive as it amounts to less than 0.5% of GDP nominal of Indonesia. While such a program would be perceived as a marked increase from current public health spending at the onset, such an investment could substantially contribute to the success of breastfeeding and substantial corresponding public health savings given that more than half of working Indonesian WRA are employed in the informal sector. Such policies should be further explored while taking into consideration realistic budget constraints and implementation capacity.

Background The economic cost of not breastfeeding in Indonesia is estimated at US$1.5–9.4 billion annually (Walters et al, 2016; Siregar et al, 2018; Walters et al, 2019), the highest in South East Asia. Half of the 33.6 million working women of reproductive age (WRA) in Indonesia are informal employees, and less than 50% exclusive breastfeed. No maternity protection entitlements are currently available for WRA working informally in Indonesia. This study aims to estimate the cost of providing maternity leave cash transfer (MCT) for WRA working in the informal sector in Indonesia.Method The costing methodology used is the adapted version of the World Bank methodology by Vilar-Compte et al (2019), following pre-set steps to estimate costs using national secondary data. We used the 2018 Indonesian National Socio-Economic Survey to estimate the number of women working informally who gave birth within the last year. The population covered, potential cash transfer’s unitary cost, the incremental coverage of the policy in terms of time and coverage, and the administrative costs were used to estimate the cost of MCT for the informal sector. Result At 100% coverage for 13 weeks of leave, the annual cost of MCT ranged from US$175million (US$152/woman) to US$669million (US$583/woman). The share of the annual costs did not exceed 0.5% of Indonesian Gross Domestic Product (GDP).Conclusion The yearly cost of providing MCT for eligible WRA working in the informal sector is significantly lower than the current annual cost of not breastfeeding in Indonesia, as computed in previous work (Walters et al, 2016; Siregar et al, 2018; Walters et al, 2019). While such a program would be perceived as a marked increase from current public health spending at the onset, such an investment could substantially contribute to the success of breastfeeding and substantial corresponding public health savings given that more than half of working Indonesian WRA are employed in the informal sector. Such policies should be further explored while taking into consideration realistic budget constraints and implementation capacity.

In this work we study a phenomenological non-gravitational interaction between dark matter and dark energy. The scenario studied in this work extends the usual interaction model proportional to the derivative of the dark component density adding to the coupling a non-linear term of the form Q = ρ /3(α + βρ). This dark sector interaction model could be interpreted as a particular case of a running vacuum model of the type Λ(H) = n0 +n1H 2 +n2H 4 in which the vacuum decays into dark matter. For a flat FRW Universe filled with dark energy, dark matter and decoupled baryonic matter and radiation we calculate the energy density evolution equations of the dark sector and solve them. The different sign combinations of the two parameters of the model show clear qualitative different cosmological scenarios, from basic cosmological insights we discard some of them. The linear scalar perturbation equations of the dark matter were calculated. Using the CAMB code we calculate the CMB and matter power spectra for some values of the parameters α and β and compare it with ΛCDM. The model modify mainly the lower multipoles of the CMB power spectrum remaining almost the same the high ones. The matter power spectrum for low wave numbers is not modified by the interaction but after the maximum it is clearly different. Using observational data from Planck, and various galaxy surveys we obtain the constraints of the parameters, the best fit values obtained are the combinations α = (3.7 ± 7) × 10 −4 , −(1.5 × 10 −5 eV −1 ) 4 β < (0.07eV −1 ) 4 . * Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México. josue@fis.unam.mx