A new magnitude--redshift relation based on SNe Ia

TL;DR

This work introduces a two-parameter empirical magnitude–redshift relation for Type Ia supernovae, yielding a simple yet accurate fit to SN Ia Hubble diagrams up to at least $z\approx1.1$ via $m(z)=\mathcal{M}+bz+5\log(z(1+z))$ and $d_L(z)=\frac{c z}{H_0}(1+z)10^{bz/5}$. The authors compare this relation with ΛCDM, flat ΛCDM, flat wCDM, and Padé cosmography, showing competitive or superior information criteria performance and robustness to the absence of low-$z$ data. They derive consistent $q_0$ values implying acceleration and obtain $H_0=73.4\pm1.0$ km s$^{-1}$ Mpc$^{-1}$ when combined with Cepheid distances. The empirical relation remains valid across eight deep-field regions, shows no significant cosmic anisotropy, and yields compatible results for DES-Dovekie and Amalgame samples, with extensions to BAO and cosmic chronometers under $\Omega_k=0$. Overall, this simple parametrization provides a practical, model-agnostic tool for SN-based cosmology and isotropy tests, with potential for validation by future deep-field surveys (JWST, LSST, CSST, Roman).

Abstract

We present a new empirical relation between the standardized magnitude ($m$) of Type Ia supernovae (SNe Ia) and redshift ($z$). Using Pantheon+ and DES-SN5YR, we find a negative linear correlation between $m-5\log(z(1+z))$ and $z$, implying that their magnitude--redshift relation can be parametrized with just two parameters: an intercept $\mathcal{M}$ and a slope $b$. This relation corresponds to the luminosity distance $d_L(z)=c\,H_0^{-1}z(1+z)10^{bz/5}$ and is valid up to at least $z\simeq1.1$. It outperforms the $Λ$CDM and flat $w$CDM models and the (2,1) Padé approximant for $d_L(z)$, and performs comparably to the flat $Λ$CDM model and the (2,1) Padé($j_0=1$) model of Hu et al. Furthermore, the relation is stable in the absence of low-$z$ SNe, making it suitable for fitting Hubble diagrams of SNe Ia without the need to add a low-$z$ sample. In deep fields in particular, assuming that the large-scale density is independent of the comoving radial coordinate, $b\propto q_0+1$. We fit the empirical relation to SN data in eight deep-field regions and find that their fitted $\mathcal{M}$ and $b$ parameters are consistent within $1.6\,σ$, in agreement with isotropy. The inferred $q_0$ values, ranging from $-0.6$ to $-0.4$, are consistent within $1.5\,σ$ and significantly lower than zero, indicating statistically consistent cosmic acceleration across all eight regions. We apply the empirical relation to the DES-Dovekie and Amalgame SN samples, finding $b$ values consistent with those from DES-SN5YR and Pantheon+. Finally, using the empirical relation in the hemispheric comparison method applied to Pantheon+ up to $z=1.1$, we find no evidence for anisotropies in $\mathcal{M}$ and $b$.

Figures (9)

• Figure 1: Sky distribution of Pantheon+ and DES-SN5YR SNe. Dashed lines indicate the Galactic equator. Plus (cross) symbols mark the north (south) Galactic pole. Circular and triangular regions are also shown.
• Figure 2: $m-5\log(z(1+z))$ versus $z$ for Pantheon+ and DES-SN5YR. Binned data (blue squares) with error bars are shown for visualization purposes only, both here and throughout the paper.
• Figure 3: Confidence contours at the 68.27% and 95.45% levels for the parameters of the empirical relation, obtained analytically and with emcee.
• Figure 4: DES-SN5YR and Pantheon+ Hubble diagrams, along with the best fits for the $m(z)$ relations. The lower panels show the residuals relative to the empirical relation.
• Figure 5: $q_0$ versus $\mathcal{M}$ for the $m(z)$ relations fitted to DES-SN5YR and Pantheon+ ($z<1.121$) with different $z_\mathrm{min}$. Solid curves are the 68.27% confidence contours for $z_\mathrm{min}=0.01$, while the contour of the empirical relation is shown as a shaded region in each panel for comparison.