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 $$\Lambda $$ Λ CDM. On the theoretical side, recent devoted studies have shown that the properly renormalized $$\rho _{\textrm{vac}}$$ ρ 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 $$\rho _{\textrm{vac}}$$ ρ vac with M turns into cosmic evolution with the Hubble rate, H. As a result the ‘cosmological constant’ $$\Lambda $$ Λ appears in our framework as the nearly sustained value of $$8\pi G(H)\rho _{\textrm{vac}}(H)$$ 8 π G ( H ) ρ 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 $$\delta \rho _\textrm{vac}(H)\sim \nu _{\textrm{eff}}\, m_{\textrm{Pl}}^2 \left( H^2-H_0^2 \right) \ (|\nu _{\textrm{eff}}|\ll 1)$$ δ ρ vac ( H ) ∼ ν eff m Pl 2 H 2 - H 0 2 ( | ν eff | ≪ 1 ) . The coefficient $$\nu _{\textrm{eff}}$$ ν 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 $$\mathcal{O}(H^{6})$$ 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.

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We have found some analytical cosmological solutions to MOdified Gravity (MOG). These solutions describe different evolutionary epochs of an isotropic and homogeneous universe. During each epoch, the evolution of cosmological perturbation is studied in the Newtonian framework and compared with the corresponding results of GR

We have considered the perturbation equations governing the growth of fluctuations during inflation in generalized scalar tensor theory f (R, φ). We have found that the scalar metric perturbations at very early times are negligible compared to the scalar field perturbation, just like general relativity. At sufficiently early times, when the physical momentum of perturbation mode, q/a is much larger than the Hubble parameter H , i.e. q/a H , we have obtained the metric and scalar field perturbation in the form of WKB solutions up to an undetermined coefficient. Then we have quantized the scalar fluctuations and expanded the metric and the scalar field perturbations with the help of annihilation and creation operators of the scalar field perturbation. The standard commutation relations of annihilation and creation operators fix the unknown coefficient. Going over to the gauge invariant quantities which are conserved beyond the horizon, we have obtained the initial condition of the generalized Mukhanov-Sasaki equation. Then a similar procedure is performed for the case of tensor metric perturbation. As an example of the generalized Mukhanov-Sasaki equation and its initial condition, we have proposed a power-law functional form as f (R, φ) = f 0 R m φ n and obtained an exact inflationary solution. In this background, then we have discussed how the scalar and tensor fluctuations grow.

We locally classify all [Formula: see text]-invariant four-dimensional pseudo-Finsler Berwald structures. These are Finslerian geometries which are closest to (spatially, or [Formula: see text])-spherically symmetric pseudo-Riemannian ones — and serve as ansatz to find solutions of Finsler gravity equations which generalize the Einstein equations. We find that there exist five classes of non-pseudo-Riemannian (i.e. non-quadratic in the velocities) [Formula: see text]-spherically symmetric pseudo-Finsler Berwald functions, which have either a heavily constrained dependence on the velocities, or, up to a suitable choice of the tangent bundle coordinates, no dependence at all on the “time” and “radial” coordinates.