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We reply to the criticisms moved in [1] against our results presented in [2]. In particular, we show that our fractal model has none of the problems claimed in [1]. The latters can be addressed to the overlooked nonlinear behaviour of the Einstein's equations., Comment: 3 pages, 1 figure

We study the so-called Gravitational Wave luminosity distance-redshift relation $d_L^{\,GW}(z)$ during cosmological eras driven by non-perfect fluids. In particular, we show that the presence of a shear viscosity in the energy momentum tensor turns out to be the most relevant effect. Within this scenario, a constant shear viscosity imprints the gravitational wave propagation through a friction term $\delta(z)$ with a uniquely given redshift dependence. This peculiar evolution predicts a specific shape for the ratio $d_{L}^{GW}/d_{L}^{EM}$ which tends to a constant value when the sources are at $z\gtrsim 1$, whereas scales linearly with the shear viscosity at lower redshifts, regardless of the value of $\Omega_{m0}$. According to our final discussion, the predicted redshift dependence $\delta(z)$ provided by a shear viscosity could be tested by upcoming surveys of multi-messenger sources against analogous scenarios provided by some widely studied theories of modified gravity., Comment: 19 pages, 4 figures. References and clarifications added. Conclusions unchanged. Accepted for publication in JCAP

We study the so-called Gravitational Wave luminosity distance-redshift relation $d_L^{\,GW}(z)$ during cosmological eras driven by non-perfect fluids. In particular, we show that the presence of a shear viscosity in the energy momentum tensor turns out to be the most relevant effect. Within this scenario, a constant shear viscosity imprints the gravitational wave propagation through a friction term $\delta(z)$ with a uniquely given redshift dependence. This peculiar evolution predicts a specific shape for the ratio $d_{L}^{GW}/d_{L}^{EM}$ which tends to a constant value when the sources are at $z\gtrsim 1$, whereas scales linearly with the shear viscosity at lower redshifts, regardless of the value of $\Omega_{m0}$. According to our final discussion, the predicted redshift dependence $\delta(z)$ provided by a shear viscosity could be tested by upcoming surveys of multi-messenger sources against analogous scenarios provided by some widely studied theories of modified gravity., Comment: 15 pages, 4 figures

We study the so-called Gravitational Wave luminosity distance-redshift relation $d_L^{\,GW}(z)$ during cosmological eras driven by non-perfect fluids. In particular, we show that the presence of a shear viscosity in the energy momentum tensor turns out to be the most relevant effect. Within this scenario, a constant shear viscosity imprints the gravitational wave propagation through a friction term $\delta(z)$ with a uniquely given redshift dependence. This peculiar evolution predicts a specific shape for the ratio $d_{L}^{GW}/d_{L}^{EM}$ which tends to a constant value when the sources are at $z\gtrsim 1$, whereas scales linearly with the shear viscosity at lower redshifts, regardless of the value of $\Omega_{m0}$. According to our final discussion, the predicted redshift dependence $\delta(z)$ provided by a shear viscosity could be tested by upcoming surveys of multi-messenger sources against analogous scenarios provided by some widely studied theories of modified gravity., Comment: 19 pages, 4 figures. References and clarifications added. Conclusions unchanged. Accepted for publication in JCAP

We study the so-called Gravitational Wave luminosity distance-redshift relation $d_L^{\,GW}(z)$ during cosmological eras driven by non-perfect fluids. In particular, we show that the presence of a shear viscosity in the energy momentum tensor turns out to be the most relevant effect. Within this scenario, a constant shear viscosity imprints the gravitational wave propagation through a friction term $\delta(z)$ with a uniquely given redshift dependence. This peculiar evolution predicts a specific shape for the ratio $d_{L}^{GW}/d_{L}^{EM}$ which tends to a constant value when the sources are at $z\gtrsim 1$, whereas scales linearly with the shear viscosity at lower redshifts, regardless of the value of $\Omega_{m0}$. According to our final discussion, the predicted redshift dependence $\delta(z)$ provided by a shear viscosity could be tested by upcoming surveys of multi-messenger sources against analogous scenarios provided by some widely studied theories of modified gravity., Comment: 19 pages, 4 figures. References and clarifications added. Conclusions unchanged. Accepted for publication in JCAP

We consider the simplest anisotropic generalization, as a correction, to the standard $\Lambda$CDM model, by replacing the spatially flat Robertson-Walker metric by the Bianchi type-I metric, which brings in a new term $\Omega_{\sigma 0}a^{-6}$ (mimicking the stiff fluid) in the average expansion rate $H(a)$ of the Universe. From Hubble and Pantheon data, relevant to the late Universe ($z\lesssim 2.4$), we obtain the constraint $\Omega_{\sigma0}\lesssim10^{-3}$, in line with the model-independent constraints. When the baryonic acoustic oscillations and cosmic microwave background (CMB) data are included, the constraint improves by 12 orders of magnitude, i.e., $\Omega_{\sigma0}\lesssim10^{-15}$. We find that this constraint could alter neither the matter-radiation equality redshift nor the peak of the matter perturbations. Demanding that the expansion anisotropy has no significant effect on the standard big bang nucleosynthesis (BBN), we find the constraint $\Omega_{\sigma0}\lesssim10^{-23}$. We show explicitly that the constraint from BBN renders the expansion anisotropy irrelevant to make a significant change in the CMB quadrupole temperature, whereas the constraint from the cosmological data in our model provides the temperature change up to $\sim11\, \rm mK$, though it is much beyond the CMB quadrupole temperature., Comment: 13 pages, 4 figures, 5 tables; Matches the published version in Physical Review D

We consider the simplest anisotropic generalization, as a correction, to the standard $\Lambda$CDM model, by replacing the spatially flat Robertson-Walker metric by the Bianchi type-I metric, which brings in a new term $\Omega_{\sigma 0}a^{-6}$ (mimicking the stiff fluid) in the average expansion rate $H(a)$ of the Universe. From Hubble and Pantheon data, relevant to the late Universe ($z\lesssim 2.4$), we obtain the constraint $\Omega_{\sigma0}\lesssim10^{-3}$, in line with the model-independent constraints. When the baryonic acoustic oscillations and cosmic microwave background (CMB) data are included, the constraint improves by 12 orders of magnitude, i.e., $\Omega_{\sigma0}\lesssim10^{-15}$. We find that this constraint could alter neither the matter-radiation equality redshift nor the peak of the matter perturbations. Demanding that the expansion anisotropy has no significant effect on the standard big bang nucleosynthesis (BBN), we find the constraint $\Omega_{\sigma0}\lesssim10^{-23}$. We show explicitly that the constraint from BBN renders the expansion anisotropy irrelevant to make a significant change in the CMB quadrupole temperature, whereas the constraint from the cosmological data in our model provides the temperature change up to $\sim11\, \rm mK$, though it is much beyond the CMB quadrupole temperature., Comment: 13 pages, 4 figures, 5 tables; Matches the published version in Physical Review D

In this paper we attempt to answer to the question: can cosmic acceleration of the Universe have a fractal solution? We give an exact solution of a Lema\^itre-Tolman-Bondi (LTB) Universe based on the assumption that such a smooth metric is able to describe, on average, a fractal distribution of matter. While the LTB model has a center, we speculate that, when the fractal dimension is not very different from the space dimension, this metric applies to any point of the fractal structure when chosen as center so that, on average, there is not any special point or direction. We examine the observed magnitude-redshift relation of type Ia supernovae (SNe Ia), showing that the apparent acceleration of the cosmic expansion can be explained as a consequence of the fractal distribution of matter when the corresponding space-time metric is modeled as a smooth LTB one and if the fractal dimension on scales of a few hundreds Mpc is $D=2.9 \pm 0.02$., Comment: 6 pages, 4 figures, accepted for publication in Classical and Quantum Gravity

In order to study if the anisotropy of the spacetime may induce differences in the cosmic shear or in deceleration parameter we investigate in Bianchi I Universe the most general expression for the cosmic shear and we study the connections between deceleration parameter and cosmic shear. We also consider a shear parametric approach to measure the transition from decelerating to an accelerating Universe. We also study the connection between jerk parameter and ellipsoidal Universe., Comment: 13 pages, accepted for publication in European Physical Journal Plus

In this paper we attempt to answer to the question: can cosmic acceleration of the Universe have a fractal solution? We give an exact solution of a Lema\^itre-Tolman-Bondi (LTB) Universe based on the assumption that such a smooth metric is able to describe, on average, a fractal distribution of matter. While the LTB model has a center, we speculate that, when the fractal dimension is not very different from the space dimension, this metric applies to any point of the fractal structure when chosen as center so that, on average, there is not any special point or direction. We examine the observed magnitude-redshift relation of type Ia supernovae (SNe Ia), showing that the apparent acceleration of the cosmic expansion can be explained as a consequence of the fractal distribution of matter when the corresponding space-time metric is modeled as a smooth LTB one and if the fractal dimension on scales of a few hundreds Mpc is $D=2.9 \pm 0.02$., Comment: 6 pages, 4 figures, accepted for publication in Classical and Quantum Gravity

In order to study if the anisotropy of the spacetime may induce differences in the cosmic shear or in deceleration parameter we investigate in Bianchi I Universe the most general expression for the cosmic shear and we study the connections between deceleration parameter and cosmic shear. We also consider a shear parametric approach to measure the transition from decelerating to an accelerating Universe. We also study the connection between jerk parameter and ellipsoidal Universe., Comment: 13 pages, accepted for publication in European Physical Journal Plus

We study the corrections to the fine-structure constant from the generalized uncertainty principle in the spacetime of a domain wall. We also calculate the corrections to the standard formula to the energy of the electron in the hydrogen atom to the ground state, in the case of spacetime of a domain wall and generalized uncertainty principle. The results generalize the cases known in literature., Comment: 8 pages. Accepted for publication in International Journal of Mathematics and Mathematical Sciences

We study the corrections to the fine-structure constant from the generalized uncertainty principle in the spacetime of a domain wall. We also calculate the corrections to the standard formula to the energy of the electron in the hydrogen atom to the ground state, in the case of spacetime of a domain wall and generalized uncertainty principle. The results generalize the cases known in literature., Comment: 8 pages. Accepted for publication in International Journal of Mathematics and Mathematical Sciences

Within the domain wall-mediated electroweak baryogenesis, we study fermion scattering off a CP-violating wall in the background of an uniform magnetic field. In particular, we calculate the asymmetry between the reflection coefficients for right-handed and left-handed chiral fermions, \Delta R = R_{R \to L} - R_{L \to R}, which is of relevance to non local baryogenesis mechanisms., Comment: replaced with revised conclusion version, to be published in JCAP

Within the domain wall-mediated electroweak baryogenesis, we study fermion scattering off a CP-violating wall in the background of an uniform magnetic field. In particular, we calculate the asymmetry between the reflection coefficients for right-handed and left-handed chiral fermions, \Delta R = R_{R \to L} - R_{L \to R}, which is of relevance to non local baryogenesis mechanisms., Comment: replaced with revised conclusion version, to be published in JCAP

Within the domain wall-mediated electroweak baryogenesis, we study fermion scattering off a CP-violating wall in the background of an uniform magnetic field. In particular, we calculate the asymmetry between the reflection coefficients for right-handed and left-handed chiral fermions, \Delta R = R_{R \to L} - R_{L \to R}, which is of relevance to non local baryogenesis mechanisms., Comment: replaced with revised conclusion version, to be published in JCAP

Within the domain wall-mediated electroweak baryogenesis, we study fermion scattering off a CP-violating wall in the background of an uniform magnetic field. In particular, we calculate the asymmetry between the reflection coefficients for right-handed and left-handed chiral fermions, \Delta R = R_{R \to L} - R_{L \to R}, which is of relevance to non local baryogenesis mechanisms., Comment: replaced with revised conclusion version, to be published in JCAP

The spontaneous generation of uniform magnetic condensate in $QED_3$ gives rise to ferromagnetic domain walls at the electroweak phase transition. These ferromagnetic domain walls are caracterized by vanishing effective surface energy density avoiding, thus, the domain wall problem. Moreover we find that the domain walls generate a magnetic field $B \simeq 10^{24} Gauss$ at the electroweak scale which account for the seed field in the so called dynamo mechanism for the cosmological primordial magnetic field. We find that the annihilation processes of walls with size $R \simeq 10^5 Km$ could release an energy of order $10^{52} erg$ indicating the invisible ferromagnetic walls as possible compact sources of Gamma Ray Bursts., Comment: LaTeX, 8 pages, 1 figure

The spontaneous generation of uniform magnetic condensate in $QED_3$ gives rise to ferromagnetic domain walls at the electroweak phase transition. These ferromagnetic domain walls are caracterized by vanishing effective surface energy density avoiding, thus, the domain wall problem. Moreover we find that the domain walls generate a magnetic field $B \simeq 10^{24} Gauss$ at the electroweak scale which account for the seed field in the so called dynamo mechanism for the cosmological primordial magnetic field. We find that the annihilation processes of walls with size $R \simeq 10^5 Km$ could release an energy of order $10^{52} erg$ indicating the invisible ferromagnetic walls as possible compact sources of Gamma Ray Bursts., Comment: LaTeX, 8 pages, 1 figure

The spontaneous generation of uniform magnetic condensate in $QED_3$ gives rise to ferromagnetic domain walls at the electroweak phase transition. These ferromagnetic domain walls are caracterized by vanishing effective surface energy density avoiding, thus, the domain wall problem. Moreover we find that the domain walls generate a magnetic field $B \simeq 10^{24} Gauss$ at the electroweak scale which account for the seed field in the so called dynamo mechanism for the cosmological primordial magnetic field. We find that the annihilation processes of walls with size $R \simeq 10^5 Km$ could release an energy of order $10^{52} erg$ indicating the invisible ferromagnetic walls as possible compact sources of Gamma Ray Bursts., Comment: LaTeX, 8 pages, 1 figure

The spontaneous generation of uniform magnetic condensate in $QED_3$ gives rise to ferromagnetic domain walls at the electroweak phase transition. These ferromagnetic domain walls are caracterized by vanishing effective surface energy density avoiding, thus, the domain wall problem. Moreover we find that the domain walls generate a magnetic field $B \simeq 10^{24} Gauss$ at the electroweak scale which account for the seed field in the so called dynamo mechanism for the cosmological primordial magnetic field. We find that the annihilation processes of walls with size $R \simeq 10^5 Km$ could release an energy of order $10^{52} erg$ indicating the invisible ferromagnetic walls as possible compact sources of Gamma Ray Bursts., Comment: LaTeX, 8 pages, 1 figure

We investigate the effects of thermal fluctuations on the spontaneous magnetic condensate in three dimensional QED coupled with P-odd Dirac fermions. Our results show that the phenomenon of the spontaneous generation of the constant background magnetic field survives to the thermal corrections even at infinite temperature. We also study the thermal corrections to the fermionic condensate in presence of the magnetic field., Comment: Final version, to appear in Phys. Lett. B

We investigate the effects of thermal fluctuations on the spontaneous magnetic condensate in three dimensional QED coupled with P-odd Dirac fermions. Our results show that the phenomenon of the spontaneous generation of the constant background magnetic field survives to the thermal corrections even at infinite temperature. We also study the thermal corrections to the fermionic condensate in presence of the magnetic field., Comment: Final version, to appear in Phys. Lett. B

The perturbation theory with a variational basis is constructed and analyzed.The generalized Gaussian effective potential is introduced and evaluated up to the second order for selfinteracting scalar fields in one and two spatial dimensions. The problem of the renormalization of the mass is discussed in details. Thermal corrections are incorporated. The comparison between the finite temperature generalized Gaussian effective potential and the finite temperature effective potential is critically analyzed. The phenomenon of the restoration at high temperature of the symmetry broken at zero temperature is discussed., Comment: RevTex, 49 pages, 16 eps figures

The perturbation theory with a variational basis is constructed and analyzed.The generalized Gaussian effective potential is introduced and evaluated up to the second order for selfinteracting scalar fields in one and two spatial dimensions. The problem of the renormalization of the mass is discussed in details. Thermal corrections are incorporated. The comparison between the finite temperature generalized Gaussian effective potential and the finite temperature effective potential is critically analyzed. The phenomenon of the restoration at high temperature of the symmetry broken at zero temperature is discussed., Comment: RevTex, 49 pages, 16 eps figures

We evaluate the thermal corrections to the generalized Gaussian effective potential. We carry out the calculations of the lowest order corrections in the case of self-interacting scalar fields in one and two spatial dimensions, and study the restoration of the symmetry at high temperatures., Comment: 11 pages + 3 figures, postscript file

We discuss the finite temperature generalized Gaussian effective potential. We put out a very simple relation between the thermal corrections to the generalizedGaussian effective potential and those of the effective potential. We evaluate explicitly the second order thermal corrections in the case of the selfinteracting scalar field in one spatial dimension., Comment: 13 pages, uuencoded compressed (using GNU's gzip) tar file containing 1 TeX file, 3 Postscript figures

We discuss the finite temperature generalized Gaussian effective potential. We put out a very simple relation between the thermal corrections to the generalizedGaussian effective potential and those of the effective potential. We evaluate explicitly the second order thermal corrections in the case of the selfinteracting scalar field in one spatial dimension., Comment: 13 pages, uuencoded compressed (using GNU's gzip) tar file containing 1 TeX file, 3 Postscript figures

We evaluate the thermal corrections to the generalized Gaussian effective potential. We carry out the calculations of the lowest order corrections in the case of self-interacting scalar fields in one and two spatial dimensions, and study the restoration of the symmetry at high temperatures., Comment: 11 pages + 3 figures, postscript file