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In this paper we study the effects of torsion of space-time in the expansion of the universe as a candidate to dark energy. The analysis is done by reconstructing the torsion function along cosmic evolution by using observational data of Supernovae type Ia and Hubble parameter measurements. We have used a kinematic model for the parameterization of the comoving distance and the Hubble parameter, then the free parameters of the models are constrained by observational data. The reconstruction of the torsion function is obtained directly from the data, using the kinematic parameterizations, and the values for the Hubble parameter and the deceleration parameter are in good agreement to the standard model estimates., Comment: 19 pages, 6 figures

In the present work, we apply consistency relation tests to several cosmological models, including the flat and non-flat $\Lambda$CDM models, as well as the flat XCDM model. The analysis uses a non-parametric Gaussian Processes method to reconstruct various cosmological quantities of interest, such as the Hubble parameter $H(z)$ and its derivatives from $H(z)$ data, as well as the comoving distance and its derivatives from SNe Ia data. We construct consistency relations from these quantities which should be valid only in the context of each model and test them with the current data. We were able to find a general method of constructing such consistency relations in the context of $H(z)$ reconstruction. In the case of comoving distance reconstruction, there were not a general method of constructing such relations and this work had to write an specific consistency relation for each model. From $H(z)$ data, we have analyzed consistency relations for all the three above mentioned models, while for SNe Ia data we have analyzed consistency relations only for flat and non-flat $\Lambda$CDM models. Concerning the flat $\Lambda$CDM model, some inconsistency was found, at more than $2\sigma$ c.l., with the $H(z)$ data in the interval $1.8\lesssim z\lesssim2.4$, while the other models were all consistent at this c.l. Concerning the SNe Ia data, the flat $\Lambda$CDM model was consistent in the $0

The possibility of an expanding decelerating Universe in the distant future is investigated in the context of a quintessence scalar field cosmology. Such a conceivable evolution is tested against SNe Ia and $H(z)$ cosmic chronometers data, and also through a model independent method based on Gaussian Processes. The scalar field model is an extension of the exponential Ratra-Peebles (RP) quintessential cosmology whose potential now depends on a pair of parameters ($\alpha, \lambda)$ and predicts a decelerated expansion in the future. Different from RP approach the $\alpha$ parameter allows for a decelerating cosmology in the future thereby frustrating the inevitable evolution for a de Sitter Cosmology as predicted by the cosmic concordance model ($\Lambda$CDM). The statistical model analysis is updated with the most recent SNe Ia and $H(z)$ data thereby obtaining $H_0 = 68.6\pm3.7$ km/s/Mpc, $\Omega_{\Phi0} = 0.735^{+0.083}_{-0.069} $, $\alpha < 6.56$ and $\lambda< 0.879 $ (at $2\sigma$ c.l.). It is also found that the extended RP model allows for a future deceleration both for $H(z)$ and SNe Ia data. In the (model-independent) Gaussian Processes analysis, however, future deceleration is allowed only in the case of $H(z)$ data., Comment: 10 pages, 5 figures. Accepted for publication in Physical Review D

By using recent $H(z)$ and SNe Ia data, we reconstruct the evolution of kinematic parameters $H(z)$, $q(z)$, jerk and snap, using a model-independent, non-parametric method, namely, the Gaussian Processes. Throughout the present analysis, we have allowed for a spatial curvature prior, based on Planck 18 [1] constraints. In the case of SNe Ia, we modify a python package (GaPP) [2] in order to obtain the reconstruction of the fourth derivative of a function, thereby allowing us to obtain the snap from comoving distances. Furthermore, using a method of importance sampling, we combine $H(z)$ and SNe Ia reconstructions in order to find joint constraints for the kinematic parameters. We find for the current values of the parameters: $H_0 =67.2 \pm 6.2$ km/s/Mpc, $q_0 = -0.60^{+0.21}_{-0.18}$, $j_0=0.90^{+0.75}_{-0.65}$, $s_0=-0.57^{+0.52}_{-0.31}$ at 1$\sigma$ c.l. We find that these reconstructions are compatible with the predictions from flat $\Lambda$CDM model, at least for 2$\sigma$ confidence intervals., Comment: 18 pages, 6 figures

Large annotated datasets have been a key component in the success of deep learning. However, annotating medical images is challenging as it requires expertise and a large budget. In particular, annotating different types of cells in histopathology suffer from high inter- and intra-rater variability due to the ambiguity of the task. Under this setting, the relation between annotators' variability and model performance has received little attention. We present a large-scale study on the variability of cell annotations among 120 board-certified pathologists and how it affects the performance of a deep learning model. We propose a method to measure such variability, and by excluding those annotators with low variability, we verify the trade-off between the amount of data and its quality. We found that naively increasing the data size at the expense of inter-rater variability does not necessarily lead to better-performing models in cell detection. Instead, decreasing the inter-rater variability with the expense of decreasing dataset size increased the model performance. Furthermore, models trained from data annotated with lower inter-labeler variability outperform those from higher inter-labeler variability. These findings suggest that the evaluation of the annotators may help tackle the fundamental budget issues in the histopathology domain

The mass dimension one (MDO) fermionic field is built on a complete set of dual-helicity eigenspinors of the charge conjugation operator, which obeys the statistic of Fermi-Dirac. These spinors are a potential candidate for the description of dark matter. The redefinition of the dual structure of this object holds a local adjacent theory and satisfies complete Lorentz invariance. In this work, we investigate important aspects of the interaction of this fermion with gravity in the light of a canonical formulation in ADM formalism. We construct an action via tetrad fields using a manifold on a family of space-like surfaces $\Sigma_t$ that carries the MDO matter field, and additionally we propose a condition for the action to have a term associated with dark energy. We found the Hamiltonian and diffeomorphism constraints at the classical level for gravitational dynamics with the immersion of this material content in space-time, which leads us to the interpretation of the contribution of dark matter energy density, parallel to the lapse function of foliation and its directional flux of energy density in the hypersurface of manifold., Comment: Larevtex4, 12 pages with 0 figures

The Co-varying Physical Couplings (CPC) framework is a modified gravity set up assuming Einstein Field Equations wherein the quantities $\{G,c,\Lambda\}$ are promoted to space-time functions. Bianchi identity and the requirement of stress-energy tensor conservation entangle the possible variations of the couplings $\{G,c,\Lambda\}$, which are forced to co-vary as dictated by the General Constraint (GC). In this paper we explore a cosmological model wherein $G$, $c$ and $\Lambda$ are functions of the redshift respecting the GC of the CPC framework. We assume a linear parametrization of $\Lambda$ in terms of the scale factor $a$. We use the ansatz $\dot{G}/G = \sigma \left( \dot{c}/c \right)$ with $\sigma =$ constant to deduce the functional forms of $c=c(z)$ and $G=G(z)$. We show that this varying-$\{G,c,\Lambda\}$ model fits SNe Ia data and $H(z)$ data with $\sigma = 3$. The model parameters can be constrained to describe dark energy at the background level., Comment: 14 pages plus references and figures; v2: matches the published version

We study a cosmological model with a fermionic field which can be interpreted as a source of dark energy in the universe. Two different approaches were considered, the first one with a massless fermionic field represented by a standard wave-function and the second one where a massive field is a Grassmann variable. {The first case naturally reduces to a XCDM model with a constant equation of state parameter, while the last case reproduces a $w(z)$CDM model for a massive field}, and in the massless limit, the intrinsic grassmannian property of the field leads always to a vacuum equation of state parameter, irrespective the specific form of the potential. Both cases leads to a dark energy contribution of the fermionic sector. The models are totally compatible with recent cosmological data from Supernovae, BAO and Hubble parameter measurements. A brief study of linear evolution of density perturbations shows that some of the small scale problems related to standard model can be at least alleviated., Comment: 21 pages, 5 figures, final version published

A cosmological model in an Einstein-Cartan framework endowed with torsion is studied. For a torsion function assumed to be proportional to Hubble expansion function, namely $\phi=-\alpha H$, the contribution of torsion function as a dark matter component is studied in two different approaches. In the first one, the total matter energy density is altered by torsion coupling $\alpha$, giving rise to an effective dark matter and cosmological constant terms that reproduce quite well the flat cosmic concordance model. In the second approach, starting with just standard baryonic matter plus a cosmological constant term, it is obtained that the coupling of torsion with baryons and cosmological constant term naturally gives rise to a dark matter contribution, together a modified cosmological term. In this model the dark matter sector can be interpreted as an effective coupling of the torsion function with the ordinary baryonic matter and cosmological constant. Finally, it is shown that both models are totally compatible with recent cosmological data from Supernovae and Hubble parameter measurements., Comment: 17 pages, 4 figures

A minimal model based on the Co-varying Physical Couplings (CPC) framework for gravity is proposed. The CPC framework is based on the assumptions of a metric-compatible four-dimensional Riemannian manifold where a covariantly conserved stress-energy tensor acts as source of the field equations which are formally the same as Einstein field equations, but where the couplings $\{ G, c,\Lambda \}$ are allowed to vary simultaneously. The minimal CPC model takes $\Lambda$ as a genuine constant while $c$ and $G$ vary in an entangled way that is consistent with Bianchi identity and the aforementioned assumptions. The model is constrained using the most recent galaxy cluster gas mass fraction observational data. Our result indicates that the functions $c(z)$ and $G\left(z\right)=G_{0}\left(c/c_{0}\right)^{4}$ are compatible with constant couplings for the three different parameterizations of $c=c(z)$ adopted here., Comment: 13 pages, 2 figures. v2: new references and some comments added. v3: content matches the published version

Scalar Fields (SF) have emerged as natural candidates for dark energy as quintessential or phantom fields, as they are the main ingredient of inflation theories. Instead of assuming some form for the scalar field potential, however, this work reconstructs the SF potential directly from observational data, namely, \textbf{Hubble and SNe Ia data}. We show that two popular forms for the SF potentials, namely, the power-law and the quadratic free-field, are compatible with the reconstructions thus obtained, at least for some choices of the priors of the matter density and curvature parameters and for some redshift intervals., Comment: New analyzes and discussions included

This paper consider an universe dominated by baryonic matter, radiation and a nonminimally coupled massive scalar field under the action of a symmetry breaking potential. Inflation occurs naturally with appropriated slow-roll values. The field evolves up to late times, when it is supposed to enter a slowing varying phase, remaining at rest at the value $\phi_0$. A mechanism for coupling the scalar field to baryonic and radiation energy densities through the nonminimal coupling $\xi$ is presented, leading naturally to dark matter and dark radiation components. A cosmological constant like term is also present, acting as a dark energy at late times. All the density parameters are in good agreement to observational data of Hubble measurements plus Supernovae Type Ia data. All the desired phases of cosmic evolution appears naturally in this simple model., Comment: 9 pages, 1 figure, 1 table

In this paper, the recently proposed mass dimension one fermionic field is supposed to be responsible for the dark matter halo around galactic nuclei, through the quantum degeneracy pressure effect of the field. It will be showed that the mass-ratio relation for dwarf galaxies can be well explained for a particle dark matter mass of about $100 - 200$eV. For a large galaxy, as Milky Way, the observational data for rotation curve can be well reproduced for a particle mass of about 23eV, with the addition of other substructures., Comment: 9 pages, 3 figures

In this paper, we implement a new method to test the invariance of the speed of light ($c$) as a function of redshift, by combining the measurements of galaxy cluster gas mass fraction, $H(z)$ from cosmic chronometers, and Type-Ia supernovae (SNe Ia). In our analyses, we consider both a constant depletion factor (which corresponds to the ratio by which the cluster gas mass fraction is depleted with respect to the universal baryonic mean) and one varying with redshift. We also consider the influence of different $H_0$ estimates on our results. We look for a variation of $c$, given by $c(z)=c_0(1+c_1z)$. We find a degeneracy between our final results on $c$ variation and the assumptions on the gas mass fraction depletion factor. Most of our analyses indicate negligible variation of the speed of light., Comment: 8 pages, 7 figures , I table, Accepted for publication in JCAP

In this work, by assuming a spatially flat Universe, we have tested 8 kinematic parametrization models with $H(z)$ data from Cosmic Chronometers and SNe Ia from Pantheon compilation. Our aim is obtain the current values for the Hubble constant ($H_0$), deceleration parameter ($q_0$), jerk ($j_0$) and snap ($s_0$) parameters independently from a dynamical model. By using a Bayesian model comparison, three models are favoured: a model with the deceleration parameter ($q$) linearly dependent on the redshift, $q$ linearly dependent on the scale factor and a model with a constant jerk. The model with constant jerk is slightly favoured by this analysis, furnishing $H_0=68.8^{+3.7}_{-3.6}$ km/s/Mpc, $q_0=-0.58\pm0.13$, $j_0=1.15^{+0.56}_{-0.53}$ and $s_0=-0.25^{+0.40}_{-0.30}$. The other models are compatible with the constant jerk model, except for the snap parameter, where we have found $s_0=4.0^{+3.4}_{-3.0}$ for the model with $q$ linearly dependent on the scale factor. (All uncertainties in the Abstract correspond to 95\% c.l.)., Comment: 23 pages, 7 figures

In this letter, we implement a test of the standard law for the dark matter density evolution. For this purpose, only a flat universe and the validity of the FRW metric are assumed. A deformed dark matter density evolution law is considered, given by $\rho_c(z) \propto (1+z)^{3+\epsilon}$, and constraints on $\epsilon$ are obtained by using galaxy cluster gas mass fractions, and cosmic chronometers measurements. We find that $\epsilon =0$ within 2$\sigma$ c.l., in full agreement with other recent analyses., Comment: 7 Pages, 3 figures, and Accepted for publication in EPJC

Many new strong gravitational lensing (SGL) systems have been discovered in the last two decades with the advent of powerful new space and ground-based telescopes. The effect of the lens mass model (usually the power-law mass model) on cosmological parameters constraints has been performed recently in literature. In this paper, by using SGL systems and Supernovae type Ia observations, we explore if the power-law mass density profile ($\rho \propto r^{-\gamma}$) is consistent with the cosmic distance duality relation (CDDR), $D_L(1+z)^{-2}/D_A=\eta(z)=1$, by considering different lens mass intervals. { It has been obtained that the verification of the CDDR validity is significantly dependent on lens mass interval considered: the sub-sample with $\sigma_{ap} \geq 300$ km/s (where $\sigma_{ap}$ is the lens apparent stellar velocity dispersion) is in full agreement with the CDDR validity, the sub-sample with intermediate $\sigma_{ap}$ values ($200 \leq \sigma_{ap} < 300)$ km/s is marginally consistent with $\eta=1$ and, finally, the sub-sample with low $\sigma_{ap}$ values ($\sigma_{ap} < 200$ km/s) ruled out the CDDR validity with high statistical confidence. Therefore, if one takes the CDDR as guarantee, our results suggest that using a single density profile is not suitable to describe lens with low $\sigma_{ap}$ values and it is only an approximate description to lenses with intermediate mass interval. }, Comment: 8 pages, 4 figures, I table, Accepted for publication in the JCAP

In this paper we study a model of interacting dark energy - dark matter where the ratio between these components is not constant, changing from early to late times in such a way that the model can solve or alleviate the cosmic coincidence problem (CP). The interaction arises from an assumed relation of the form $\rho_x \propto \rho_d^\alpha$, where $\rho_x$ and $\rho_d$ are the energy densities of dark energy and dark matter components, respectively, and $\alpha$ is a free parameter. For a dark energy equation of state parameter $w=-1$ we found that, if $\alpha=0$, the standard $\Lambda$CDM model is recovered, where the coincidence problem is unsolved. For $0<\alpha<1$, the CP would be alleviated and for $\alpha\sim 1$, the CP would be solved. The dark energy component is analyzed with both $w=-1$ and $w \neq -1$. Using Supernovae type Ia and Hubble parameter data constraints, in the case $w=-1$ we find $\alpha=0.109^{+0.062}_{-0.072}$ at 68% C.L., and the CP is alleviated. For $w\neq -1$, a degeneracy arises on the $w$ - $\alpha$ plane. In order to break such degeneracy we add cosmic microwave background distance priors and baryonic acoustic oscillations data to the constraints, yielding $\alpha=-0.075\pm 0.046$ at 68% C.L.. In this case we find that the CP is not alleviated even for 2$\sigma$ interval for $\alpha$. Furthermore, this last model is discarded against flat $\Lambda$CDM according to BIC analysis., Comment: 22 pages, 6 figures, revised version, with Bayesian Information Criterion (BIC) analysis included. Version published

A viable model for inflation driven by a torsion function in a Friedmann background is presented. The scalar spectral index in the interval $0.92\lesssim n_{s}\lesssim 0.97$ is obtained in order to satisfy the initial conditions for inflation. The post inflationary phase is also studied, and the analytical solutions obtained for scale factor and energy density generalizes that ones for a matter dominated universe, indicating just a small deviation from the standard model evolution. The same kind of torsion function used also describes satisfactorily the recent acceleration of the universe, which could indicate a possible unification of different phases, apart form specific constants., Comment: 8 pages, 2 figures, submitted to EPJC

In an environmental framework, extreme values of certain spatio-temporal processes, for example wind speeds, are the main cause of severe damage in property, such as electrical networks, transport and agricultural infrastructures. Therefore, availability of accurate data on such processes is highly important in risk analysis, and in particular in producing probability maps showing the spatial distribution of damage risks. Typically, as is the case of wind speeds, data are available at few stations with many missing observations and consequently simulated data are often used to augment information, due to simulated environmental data being available at high spatial and temporal resolutions. However, simulated data often mismatch observed data, particularly at tails, therefore calibrating and bringing it in line with observed data may offer practitioners more reliable and richer data sources. Although the calibration methods that we describe in this manuscript may equally apply to other environmental variables, we describe the methods specifically with reference to wind data and its consequences. Since most damages are caused by extreme winds, it is particularly important to calibrate the right tail of simulated data based on observations. Response relationships between the extremes of simulated and observed data are by nature highly non-linear and non-Gaussian, therefore data fusion techniques available for spatial data may not be adequate for this purpose. After giving a brief description of standard calibration and data fusion methods to update simulated data based on the observed data, we propose and describe in detail a specific conditional quantile matching calibration method and show how our wind speed data can be calibrated using this method.

An accurate determination of the Hubble constant remains a puzzle in observational cosmology. The possibility of a new physics has emerged with a significant tension between the current expansion rate of our Universe measured from the cosmic microwave background by the Planck satellite and from local methods. In this paper, new tight estimates on this parameter are obtained by considering two data sets from galaxy distribution observations: galaxy cluster gas mass fractions and baryon acoustic oscillation measurements. Priors from the Big Bang nucleosynthesis (BBN) were also considered. By considering the flat $\Lambda$CDM and XCDM models, and the non-flat $\Lambda$CDM model, our main results are: $H_0=65.9^{+1.5}_{-1.5}$ km s$^{-1}$ Mpc$^{-1}$, $H_0=65.9^{+4.4}_{-4.0}$ km s$^{-1}$ Mpc$^{-1}$ and $H_0=64.3^{+ 4.5}_{- 4.4}$ km s$^{-1}$ Mpc$^{-1}$ in $2\sigma$ c.l., respectively. These estimates are in full agreement with the Planck satellite results. Our analyses in these cosmological scenarios also support a negative value for the deceleration parameter at least in 3$\sigma$ c.l.., Comment: 16 pages, 6 figues, 2 table, version accepted in JCAP

We present an efficient machine learning framework for detection and classification of nanoparticles on surfaces that are detected in the far-field with Coherent Fourier Scatterometry (CFS). We study silicon wafers contaminated with spherical polystyrene (PSL) nanoparticles (with diameters down to $\lambda/8$). Starting from the raw data, the proposed framework does the pre-processing and particle search. Further, the unsupervised clustering algorithms, such as K-means and DBSCAN, are customized to be used to define the groups of signals that are attributed to a single scatterer. Finally, the particle count versus particle size histogram is generated. The challenging cases of the high density of scatterers, noise and drift in the dataset are treated. We take advantage of the prior information on the size of the scatterers to minimize the false-detections and as a consequence, provide higher discrimination ability and more accurate particle counting. Numerical and real experiments are conducted to demonstrate the performance of the proposed search and cluster-assessment techniques. Our results illustrate that the proposed algorithm can detect surface contaminants correctly and effectively., Comment: Manuscript in preparation

This paper aims to put constraints on the parameters of the Scalar Field Dark Matter (SFDM) model, when dark matter is described by a free real scalar field filling the whole Universe, plus a cosmological constant term. By using a compilation of 51 $H(z)$ data and 1048 Supernovae data from Panteon, a lower limit for the mass of the scalar field was obtained, $m \geq 5.1\times 10^{-34} $eV and $H_0=69.5^{+2.0}_{-2.1}\text{ km s}^{-1}\text{Mpc}^{-1}$. Also, the present dark matter density parameter was obtained as $\Omega_\phi = 0.230^{+0.033}_{-0.031}$ at $2\sigma$ confidence level. The results are in good agreement to standard model of cosmology, showing that SFDM model is viable in describing the dark matter content of the universe., Comment: 15 pages, 1 table and 2 figures. arXiv admin note: substantial text overlap with arXiv:1907.01033

In this work we study and compare the features of gravitational entropy near the throat of transversable wormholes formed by exotic matter and wormholes in galactic halos. We have verified that gravitational entropy and entropy density of these wormholes in regions near their throats are indistinguishable for objects of same throat, despite the fact they are described by different metrics and by distinct energy-momentum tensors. We have found that the gravitational entropy density diverges near the throat for both cases, probably due to a non-trivial topology at this point, however allowing the interesting interpretation that a maximum flux of information can be carried through the throat of these wormholes. In addition, we have found that both are endowed with an entropic behaviour similar to Hawking-Bekenstein's entropy of non-rotating and null charge black holes., Comment: 18 pages, 3 figures

This document complements and completes what was submitted last year to PAC45 as an update to the proposal PR12-16-001 "Dark matter search in a Beam-Dump eXperiment (BDX)" at Jefferson Lab submitted to JLab-PAC44 in 2016. Following the suggestions contained in the PAC45 report, in coordination with the lab, we ran a test to assess the beam-related backgrounds and validate the simulation framework used to design the BDX experiment. Using a common Monte Carlo framework for the test and the proposed experiment, we optimized the selection cuts to maximize the reach considering simultaneously the signal, cosmic-ray background (assessed in Catania test with BDX-Proto) and beam-related backgrounds (irreducible NC and CC neutrino interactions as determined by simulation). Our results confirmed what was presented in the original proposal: with 285 days of a parasitic run at 65 $\mu$A (corresponding to $10^{22}$ EOT) the BDX experiment will lower the exclusion limits in the case of no signal by one to two orders of magnitude in the parameter space of dark-matter coupling versus mass.

This paper aims to put constraints on the transition redshift $z_t$, which determines the onset of cosmic acceleration, in cosmological-model independent frameworks. In order to do that, we use the non-parametric Gaussian Process method with $H(z)$ and SNe Ia data. The deceleration parameter reconstruction from $H(z)$ data yields $z_t=0.59^{+0.12}_{-0.11}$. The reconstruction from SNe Ia data assumes spatial flatness and yields $z_t=0.683^{+0.11}_{-0.082}$. These results were found with a Gaussian kernel and we show that they are consistent with two other kernel choices., Comment: 12 figures, 19 pages

A Friedmann like cosmological model in Einstein-Cartan framework is studied when the torsion function is assumed to be proportional to a single $\phi(t)$ function coming just from the spin vector contribution of ordinary matter. By analysing four different types of torsion function written in terms of one, two and three free parameters, we found that a model with $\phi(t)=- \alpha H(t) \big({\rho_{m}(t)}/{\rho_{0c}}\big)^n$ is totally compatible with recent cosmological data, where $\alpha$ and $n$ are free parameters to be constrained from observations, $\rho_m$ is the matter energy density and $\rho_{0c}$ the critical density. The recent accelerated phase of expansion of the universe is correctly reproduced by the contribution coming from torsion function, with a deceleration parameter indicating a transition redshift of about $0.65$., Comment: 17 pages, 6 figures

In this paper, we propose a new test to the cosmic distance duality relation (CDDR), $D_L=D_A(1+z)^2$, where $D_L$ and $D_A$ are the luminosity and angular diameter distances, respectively. The data used correspond to 61 Type Ia Supernova luminosity distances and $Y_{SZE}-Y_X$ measurements of 61 galaxy clusters obtained by the {\it Planck} mission and the deep XMM-Newton X-ray data, where $Y_{SZE}$ is the integrated comptonization parameter obtained via Sunyaev-Zel'dovich effect observations and $Y_X$ is the X-ray counterpart. More precisely, we use the $Y_{SZE}D_{A}^{2}/C_{XSZE}Y_X$ scaling-relation and a deformed CDDR, such as $D_L/D_A(1+z)^2=\eta(z)$, to verify if $\eta(z)$ is compatible with the unity. Two $\eta(z)$ functions are used, namely, $\eta(z)=1+\eta_0 z$ and $\eta(z)=1+\eta_0 z /(1+z)$. { We obtain that the CDDR validity ($\eta_0=0$) is verified within $\approx 1.5\sigma$ c.l. for both $\eta(z)$ functions.}., Comment: 7 pages, two figures, I table, Version accepted for publication in the JCAP