We propose a novel deep learning tool in order to study the evolution of dark energy models. The aim is to combine a training of Recurrent Neural Networks (RNN) and of Bayesian Neural Networks (BNN). The first one is capable of learning complex sequential information to classify objects like supernovae and use the light-curves directly to learn information from the sequence of observations. Since RNN is not capable to calculate the uncertainties, BNN emerges as a solution for problems in deep learning like, for example, the overfitting. For the trainings we use measurements of the distance modulus µ(z), such as those provided by Pantheon Supernovae Type Ia. In view of our results, the reported approach turns out to be a first promising step on how we can train a neural network for specific cosmological data. It is worth stressing that the technique allows to reduce the computational load of expensive codes for dark energy models and probe the necessity of modified dark energy models at higher redshift than that reported by current supernovae astrophysical samples. PACS numbers: 98.80.-k, 98.80.Es, 07.05.Mh
There is a strong discrepancy between the value of the Hubble parameter $$H_0^P$$ H 0 P obtained from large scale observations such as the Planck mission, and the small scale value $$H_0^R$$ H 0 R , obtained from low redshift supernovae (SNe). The value of the absolute magnitude $$M^{Hom}$$ M Hom used as prior in analyzing observational data is obtained from low-redshift SNe, assuming a homogeneous Universe, but the distance of the anchors used to calibrate the SNe to obtain M would be affected by a local inhomogeneity, making it inconsistent to test the Copernican principle using $$M^{Hom}$$ M Hom , since M estimation itself is affected by local inhomogeneities. We perform an analysis of the luminosity distance of low redshift SNe, using different values of M, $$\{M^P,M^R\}$$ { M P , M R } , corresponding to different values of $$H_0$$ H 0 , $$\{H_0^P,H_0^R\}$$ { H 0 P , H 0 R } , obtained from the model independent consistency relation between $$H_0$$ H 0 and M which can be derived from the definition of the distance modulus. We find that the value of M can strongly affect the evidence of a local inhomogeneity. We analyze data from the Pantheon catalog, finding no significant statistical evidence of a local inhomogeneity using the parameters $$\{M^R,H_0^R\}$$ { M R , H 0 R } , confirming previous studies, while with $$\{M^P,H_0^P\}$$ { M P , H 0 P } we find evidence of a small local void, which causes an overestimation of $$M^R$$ M R with respect to $$M^P$$ M P . An inhomogeneous model with the parameters $$\{M^P,H_0^P\}$$ { M P , H 0 P } fits the data better than a homogeneous model with $$\{M^R,H_0^R\}$$ { M R , H 0 R } , resolving the apparent $$H_0$$ H 0 tension. Using $$\{M^P,H_0^P\}$$ { M P , H 0 P } , we obtain evidence of a local inhomogeneity with a density contrast $$-0.140 \pm 0.042 $$ - 0.140 ± 0.042 , extending up to a redshift of $$z_v =0.056 \pm 0.0002$$ z v = 0.056 ± 0.0002 , in good agreement with recent results of galaxy catalogs analysis (Wong et al. in The local hole: a galaxy under-density covering 90 mpc, 2021).
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