tokens: limits were calculated from the misfit function when all solutions with a fit at least of 99percent of the best fit were accepted. Figure 2. Inversion for optimum cosmological parameters of (a) the CDM model and (b) the opaque universe model. The colour shows the mean of the absolute distance modulus residua between the predicted model and the SN Ia data as a function of (a) the Hubble constant H 0 and the dark energy , and (b) the Hubble constant H 0 and the B -band intergalactic opacity B . Only the most accurate 552 SNe Ia with z 1.4 and an error less than 0.50 mag are used. The optimum solutions marked by the white plus signs are defined by (a) and ( 0.122 and 2.466) and (b) and ( 0.088 and 1.920). The residual Hubble plots are shown in Fig. 3 . The optimum solutions have roughly comparable misfits for both models: 0.16 for the CDM model (Fig. 3a ) and 0.17 for the model of the opaque universe (Fig. 3b ). However, as illustrated in Figs 3(c) and (d), the CDM model performs slightly better than the opaque universe model, because of a better fit of the binned residua for redshifts less than 0.6. Nevertheless, the model of the opaque universe also accounts for the observed dimming of the SNe Ia within the confidence level of 95percent (Fig. 3d ). Figure 3. Residual Hubble plots for (a, b) the individual SN Ia data and (c, d) the binned SN Ia data. (a, c) The flat CDM model with and . (b, d) The opaque universe model with and . The error bars in (c, d) show the 95percent confidence intervals. Data are taken from Suzuki etal. ( 2012 ). Importantly, the optimum solutions are sensitive to the redshift interval of the SNe Ia used in the inversion. If the optimum model of the opaque universe is searched using the SNe Ia with redshifts z 0.6 only, and the SNe Ia are grouped in the following redshift bins: z [0, 0.025, 0.05, 0.1, 0.15, 0.20, 0.25, 0.30, 0.35, 0.40, 0.45, 0.5, 0.6], we get and (Fig. 4 ). We see in Fig. 4(b) that the model fits the SN Ia measurements at z 0.6 much better than in Fig. 3(d) , but a significant discrepancy appears at higher redshifts. A similar result was obtained by Goobar etal. ( 2002 ) and Riess etal. ( 2004 ), who concluded that the discrepancy at high redshifts excludes the SN Ia dimming to be produced by intergalactic dust. Figure 4. Residual Hubble plots for (a) the individual and (b) binned SN Ia data for the opaque universe model with and . The vertical dashed line in (b) denotes the upper redshift limit of the SN Ia data considered in the inversion. The error bars in (b) show the 95percent confidence intervals. Data are taken from Suzuki etal. ( 2012 ). Note that the best estimate of H 0 for the CDM model obtained by Riess etal. ( 2016 ) using the SN Ia data is . The precision of the distance scale was further improved by a factor of 2.5 by Riess etal. ( 2018 ). This value is within the uncertainty of the estimate of H 0 for the CDM model obtained in this paper and shown in Fig. 2(a) . The accuracy of the
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