arxiv: 2601.08505 · v2 · submitted 2026-01-13 · 🌌 astro-ph.CO · gr-qc

Recognition: no theorem link

A new magnitude--redshift relation based on Type Ia supernovae

\'Osmar Rodr\'iguez , Alejandro Clocchiatti

Authors on Pith no claims yet

Pith reviewed 2026-05-16 15:04 UTC · model grok-4.3

classification 🌌 astro-ph.CO gr-qc

keywords Type Ia supernovaemagnitude-redshift relationluminosity distancecosmic accelerationisotropyHubble diagramPantheon+DES supernova

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The pith

Type Ia supernovae follow a two-parameter magnitude-redshift relation that holds to z=1.1 and matches standard models.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper identifies a linear correlation in Type Ia supernova data after subtracting 5 log of z(1+z) from the standardized magnitude, leaving a quantity that decreases steadily with redshift. This leads to a simple luminosity distance formula using only an intercept M and slope b, written as d_L(z) equals c over H0 times z times (1 plus z) times 10 to the power of b z over 5. The form fits Pantheon+ and DES-SN5YR observations at least as well as LambdaCDM or wCDM models and stays consistent when low-redshift supernovae are removed entirely. In separate deep fields the slope b connects to the deceleration parameter, producing negative q0 values that agree across patches and point to uniform acceleration with no detected directional differences.

Core claim

Using Pantheon+ and DES-SN5YR data the authors find a negative linear correlation between m minus 5 log of z(1+z) and redshift z. This correlation parametrizes the magnitude-redshift relation with just two numbers, an intercept M and slope b, which corresponds to the luminosity distance d_L(z) equals c H_0 inverse times z(1+z) times 10 to the b z over 5 and remains valid at least to z approximately 1.1. The parametrization outperforms or equals LambdaCDM and flat wCDM models in fits, performs comparably to certain Pade approximants, and remains stable without low-z supernovae. In deep fields, under the assumption that large-scale density is independent of comoving radial coordinate, the fitb

What carries the argument

The observed linear correlation between transformed magnitude m - 5 log[z(1+z)] and redshift z, which supplies the two free parameters M and b in the luminosity distance expression d_L(z) = c H_0^{-1} z(1+z) 10^{b z /5}.

If this is right

The two-parameter relation fits the supernova Hubble diagram as well as or better than LambdaCDM and flat wCDM up to z=1.1.
The relation remains stable and usable for fitting when low-redshift supernovae are excluded from the sample.
Fitted M and b values are consistent within 1.6 sigma across eight separate deep-field regions.
Inferred q0 values between -0.6 and -0.4 are consistent within 1.5 sigma and negative, indicating cosmic acceleration in every region examined.
Hemispheric comparison of Pantheon+ data shows no anisotropy in the parameters M or b.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The reduced parameter count may simplify cosmological fits in future surveys that lack complete low-redshift coverage.
If the proportionality between b and q0 +1 holds, high-redshift-only samples could yield direct deceleration measurements without full dark-energy modeling.
Consistency of parameters across fields and hemispheres supports extending the uniformity assumption to larger scales probed by next-generation telescopes.
The empirical form offers a model-independent baseline that could be tested against other distance indicators such as baryon acoustic oscillations at overlapping redshifts.

Load-bearing premise

Large-scale density is independent of comoving radial coordinate, which makes the fitted slope b proportional to q0 plus one.

What would settle it

A statistically significant departure from linearity in the plot of m - 5 log[z(1+z)] versus z when a larger supernova sample extends beyond z=1.1, or b values that differ by more than 1.6 sigma between independent deep-field regions.

Figures

Figures reproduced from arXiv: 2601.08505 by Alejandro Clocchiatti, \'Osmar Rodr\'iguez.

**Figure 1.** 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. where ΩM + ΩΛ + Ωk = 1 and w is the dark energy equation-ofstate parameter. We use the m(z) relations for ΛCDM (w = −1), flat ΛCDM (Ωk = 0, w = −1), and the phenomenological flat wCDM (Ωk = 0) models. 3.1.2. … view at source ↗

**Figure 2.** 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. & Anderson 2002), while ∆BIC < 2 and 6 < ∆BIC < 10 indicate little and strong evidence against the model, respectively (Kass & Raftery 1995). Therefore, models with both ∆AICc < 2 and ∆BIC < 2 perform comparably, yielding compara… view at source ↗

**Figure 4.** 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. model, and the Padé cosmography are less supported by the AICc and disfavoured by the BIC, reflecting the penalty associated with their additional free parameter. 4.4. Deceleration parameter and Hubble constant [PITH_FULL_IMAGE:figures/full_fig_… view at source ↗

**Figure 5.** Figure 5: q0 versus M for the m(z) relations fitted to DES-SN5YR and Pantheon+ (z < 1.121) with different zmin. Solid curves are the 68.27% confidence contours for zmin = 0.01, while the contour of the empirical relation is shown as a shaded region in each panel for comparison. Note that, although the flat wCDM constraints in the (w, ΩM) plane are highly non-Gaussian, their projection onto q0 leads to nearly symmet… view at source ↗

**Figure 6.** Figure 6: Same as [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗

**Figure 7.** Figure 7: Confidence contours (68.27%) and marginalized distributions for the parameters of the empirical relation fitted to the deep-field regions. The parameters are consistent with those from DES-SN5YR and from Pantheon+ (z < 1.121), as well as with those reported by Popovic et al. (2025) for the flat ΛCDM (ΩM = 0.330 ± 0.015), ΛCDM (ΩM = 0.279±0.057, Ωk = 0.14±0.15), and flat wCDM model (ΩM = 0.263+0.064 −0.078,… view at source ↗

**Figure 8.** Figure 8: Same as [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗

**Figure 9.** Figure 9: Sky maps of S/N(∆M) and S/N(∆b). Dots represent the SNe in Pantheon+ (z < 1.121) and circles mark the maximum S/N values. To evaluate the probabilities of obtaining S/N(∆M) ≥ 3.7 and S/N(∆b) ≥ 3.3 by chance when scanning many directions on the sky (the look-elsewhere effect; Bayer & Seljak 2020), we perform 104 simulations. In each simulation, we randomize the RA and Dec coordinates of the SNe and produce … view at source ↗

read the original 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 $\Lambda$CDM and flat $w$CDM models and the (2,1) Pad\'e approximant for $d_L(z)$, and performs comparably to the flat $\Lambda$CDM model and the (2,1) Pad\'e($j_0=1$) 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 $1.6\,\sigma$, in agreement with isotropy. The inferred $q_0$ values, ranging from $-0.6$ to $-0.4$, are consistent within $1.5\,\sigma$ 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 $b$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper spots a linear trend in transformed SN magnitudes that yields a compact two-parameter d_L form fitting the data up to z=1.1, but the step linking slope b to q0 rests on an assumption whose derivation is not shown.

read the letter

The main point is that Rodriguez and Clocchiatti report a negative linear correlation between m - 5 log(z(1+z)) and z in both Pantheon+ and DES-SN5YR. This lets them write the luminosity distance as d_L(z) = c H_0^{-1} z(1+z) 10^{b z /5} using only an intercept M and slope b, and the form holds at least to z=1.1. They also show the fit remains stable when low-z supernovae are dropped, which is the practical part for deep-field work. They then apply the same relation to eight separate deep-field regions, find the fitted M and b values consistent within 1.6 sigma, and recover q0 values between -0.6 and -0.4 that are consistent within 1.5 sigma and clearly negative. The hemispheric test on Pantheon+ up to z=1.1 likewise shows no anisotropy in either parameter. That empirical consistency check is the part that actually adds something usable. The soft spot is the physical reading of b. The authors state that if large-scale density is independent of comoving radial coordinate then b is proportional to q0 + 1, but the abstract gives no derivation of that proportionality and standard FLRW already has radial independence in comoving coordinates while density still evolves as (1+z)^3. Without the missing steps, the claim that the data show consistent acceleration across fields is harder to judge. The outperformance statements versus LambdaCDM and wCDM are also just goodness-of-fit comparisons on the same data, so they are expected for any two-parameter empirical form. This is for people who fit supernova Hubble diagrams or test isotropy in deep fields. A reader who needs a simple parametrization that works without low-z anchors will find concrete numbers and checks here. The data handling looks solid enough that the paper deserves a serious referee, though the authors will need to supply the missing link between b and q0 before the physical conclusions can be taken at face value.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes an empirical two-parameter magnitude-redshift relation for Type Ia supernovae, m - 5 log(z(1+z)) = M + b z, fitted to Pantheon+ and DES-SN5YR data. This corresponds to the luminosity distance form d_L(z) = c H_0^{-1} z(1+z) 10^{b z /5}, claimed valid to z ≃ 1.1. The relation is reported to outperform ΛCDM, flat wCDM, and a (2,1) Padé approximant in fits, to be stable without low-z samples, and to yield consistent M and b (hence q_0 ∈ [-0.6, -0.4]) across eight deep-field regions under the assumption of radially independent large-scale density, supporting isotropic acceleration. Hemispheric tests on Pantheon+ up to z=1.1 show no anisotropy.

Significance. If the empirical linearity is robust and the mapping from b to q_0 is rigorously justified, the two-parameter form would simplify SN Ia Hubble-diagram fitting (especially in deep fields lacking low-z anchors) and provide a direct, falsifiable test of isotropy and acceleration. The reported consistency of q_0 < 0 across independent deep-field patches would strengthen evidence for homogeneous acceleration, while the stability without low-z data could reduce systematic dependence on local calibrators.

major comments (2)

[deep-field analysis] Deep-field analysis section: the claim that b ∝ q_0 + 1 follows from the assumption that large-scale density is independent of the comoving radial coordinate is stated without derivation. Standard FLRW homogeneity already implies comoving radial independence, yet the physical density scales as (1+z)^3; the manuscript must show explicitly how radial independence plus the chosen d_L(z) functional form produces exactly this linear proportionality for b, otherwise the inferred q_0 range [-0.6, -0.4] and the acceleration conclusion rest on an unverified step.
[results] Results and model-comparison paragraphs: the assertion that the empirical relation outperforms ΛCDM and flat wCDM lacks quantitative details on the fitting procedure (covariance-matrix treatment, χ² definition, degrees of freedom, or information criteria such as AIC/BIC). Because M and b are extracted directly from the same supernova data used for the comparison, any reported outperformance must be shown to exceed the improvement expected from the difference in parameter count; without these metrics the performance claim cannot be evaluated.

minor comments (2)

[abstract and §2] Notation: the intercept is denoted both as M and script-M in the abstract and text; adopt a single symbol throughout and define it explicitly in terms of the distance modulus.
[data and fits] The statement that the relation is 'valid up to at least z ≃ 1.1' should be accompanied by the highest redshift actually used in the fits and by a residual plot versus z to demonstrate the absence of systematic trends beyond that range.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and will revise the manuscript to incorporate the requested clarifications and additional details.

read point-by-point responses

Referee: Deep-field analysis section: the claim that b ∝ q_0 + 1 follows from the assumption that large-scale density is independent of the comoving radial coordinate is stated without derivation. Standard FLRW homogeneity already implies comoving radial independence, yet the physical density scales as (1+z)^3; the manuscript must show explicitly how radial independence plus the chosen d_L(z) functional form produces exactly this linear proportionality for b, otherwise the inferred q_0 range [-0.6, -0.4] and the acceleration conclusion rest on an unverified step.

Authors: We thank the referee for highlighting the need for an explicit derivation. The claimed proportionality follows from matching the second-order term in the Taylor expansion of our empirical luminosity distance to the standard FLRW expansion. Specifically, d_L(z) = c H_0^{-1} z(1+z) 10^{b z /5} expands as d_L(z) ≈ c H_0^{-1} [z + (1 + (b ln(10))/5) z^2 + O(z^3)]. The standard FLRW form is d_L(z) ≈ c H_0^{-1} [z + ((1 - q_0)/2) z^2 + O(z^3)]. Equating the coefficients of z^2 gives 1 + (b ln(10))/5 = (1 - q_0)/2, hence b = -(5/(2 ln(10))) (1 + q_0), so b ∝ -(q_0 + 1). The assumption of large-scale density being independent of comoving radial coordinate is the homogeneity assumption of the FLRW metric; the (1+z)^3 scaling of physical density is already incorporated in the standard derivation of the luminosity-distance expansion. We will add this derivation to the deep-field analysis section. revision: yes
Referee: Results and model-comparison paragraphs: the assertion that the empirical relation outperforms ΛCDM and flat wCDM lacks quantitative details on the fitting procedure (covariance-matrix treatment, χ² definition, degrees of freedom, or information criteria such as AIC/BIC). Because M and b are extracted directly from the same supernova data used for the comparison, any reported outperformance must be shown to exceed the improvement expected from the difference in parameter count; without these metrics the performance claim cannot be evaluated.

Authors: We agree that quantitative details on the fitting procedure and model comparison are essential. In the revised manuscript we will expand the relevant sections to specify: the exact χ² definition and full covariance-matrix treatment for Pantheon+ and DES-SN5YR; the number of degrees of freedom for each model; and the computed AIC and BIC values for the empirical relation, ΛCDM, flat wCDM, and the Padé approximant. These information criteria will demonstrate that any improvement exceeds the penalty expected from the difference in parameter count (two parameters for the empirical model). We will also note that while M and b are fitted to the same data, the information criteria properly account for model complexity. revision: yes

Circularity Check

0 steps flagged

No significant circularity in empirical parametrization

full rationale

The paper identifies a linear correlation directly from the Pantheon+ and DES-SN5YR supernova data in the transformed quantity m-5log(z(1+z)) versus z, then defines the two-parameter form (M, b) and the corresponding d_L(z) expression as the direct mathematical consequence of that observed linearity. This is presented explicitly as an empirical finding rather than a first-principles derivation or prediction. Model comparisons consist of standard goodness-of-fit evaluations on the identical dataset, which do not constitute a reduction by construction. The proportionality b∝q0+1 is introduced under an explicit external assumption about large-scale density, without any claim that the data themselves derive or enforce it. No self-citations, uniqueness theorems, or ansatzes from prior author work appear as load-bearing steps in the provided text.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The model rests on two fitted parameters and two domain assumptions about the linearity of the transformed data and the radial independence of large-scale density; no new physical entities are introduced.

free parameters (2)

intercept M
Fitted intercept in the linear relation between m-5 log(z(1+z)) and z
slope b
Fitted slope in the linear relation between m-5 log(z(1+z)) and z; also used to infer q_0

axioms (2)

domain assumption The standardized magnitudes follow a linear trend with redshift in the variable m-5 log(z(1+z))
Directly invoked to justify the two-parameter parametrization
domain assumption Large-scale density is independent of the comoving radial coordinate
Invoked to relate the fitted slope b to the deceleration parameter q_0 + 1

pith-pipeline@v0.9.0 · 5720 in / 1471 out tokens · 54930 ms · 2026-05-16T15:04:46.203947+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

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Reference graph

Works this paper leans on

47 extracted references · 47 canonical work pages · cited by 1 Pith paper · 2 internal anchors

[1]

2025, Phys

Abdul Karim, M., Aguilar, J., Ahlen, S., et al. 2025, Phys. Rev. D, 112, 083515

work page 2025
[2]

C., Cafaro, C., Capozziello, S., Luongo, O., & Muccino, M

Alfano, A. C., Cafaro, C., Capozziello, S., Luongo, O., & Muccino, M. 2026, Journal of High Energy Astrophysics, 49, 100444

work page 2026
[3]

Baker, G. A. & Graves-Morris, P. 1996, Pade approximants (Cambridge, UK: Cambridge University Press)

work page 1996
[4]

Bayer, A. E. & Seljak, U. 2020, J. Cosmology Astropart. Phys., 2020, 009

work page 2020
[5]

2015, ApJ, 814, 7

Bochner, B., Pappas, D., & Dong, M. 2015, ApJ, 814, 7

work page 2015
[6]

2022, ApJ, 938, 110

Brout, D., Scolnic, D., Popovic, B., et al. 2022, ApJ, 938, 110

work page 2022
[7]

Burnham, K. P. & Anderson, D. R. 2002, Model selection and multimodel infer- ence: a practical information-theoretic approach (New York: Springer-Verlag)

work page 2002
[8]

M., Vincenzi, M., et al

Camilleri, R., Davis, T. M., Vincenzi, M., et al. 2024, MNRAS, 533, 2615

work page 2024
[9]

2020, MNRAS, 494, 2576

Capozziello, S., D’Agostino, R., & Luongo, O. 2020, MNRAS, 494, 2576

work page 2020
[10]

M., Kartaltepe, J

Casey, C. M., Kartaltepe, J. S., Drakos, N. E., et al. 2023, ApJ, 954, 31 Cattoën, C. & Visser, M. 2007, Classical and Quantum Gravity, 24, 5985

work page 2023
[11]

2024, ApJ, 971, 19

Clocchiatti, A., Rodríguez, Ó., Órdenes Morales, A., & Cuevas-Tapia, B. 2024, ApJ, 971, 19

work page 2024
[12]

2011, ApJS, 192, 1

Conley, A., Guy, J., Sullivan, M., et al. 2011, ApJS, 192, 1

work page 2011
[13]

DeCoursey, C., Egami, E., Pierel, J. D. R., et al. 2025, ApJ, 979, 250

work page 2025
[14]

Earp, S. W. F., Debattista, V . P., Macciò, A. V ., et al. 2019, MNRAS, 488, 5728

work page 2019
[15]

2025, Philosophical Transactions of the Royal Society of London Series A, 383, 20240022

Efstathiou, G. 2025, Philosophical Transactions of the Royal Society of London Series A, 383, 20240022

work page 2025
[16]

2008, General Relativity and Gravitation, 40, 451

Enqvist, K. 2008, General Relativity and Gravitation, 40, 451

work page 2008
[17]

W., Lang, D., & Goodman, J

Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125, 306

work page 2013
[18]

& Rubin, D

Gelman, A. & Rubin, D. B. 1992, Statistical Science, 7, 457 Górski, K. M., Hivon, E., Banday, A. J., et al. 2005, ApJ, 622, 759

work page 1992
[19]

R., et al

Gris, P., Awan, H., Becker, M. R., et al. 2024, ApJS, 275, 21

work page 2024
[20]

2023, arXiv e-prints, arXiv:2307.02670

Hounsell, R., Scolnic, D., Brout, D., et al. 2023, arXiv e-prints, arXiv:2307.02670

work page arXiv 2023
[21]

Hu, J. P. & Wang, F. Y . 2022, A&A, 661, A71

work page 2022
[22]

2025, Science China Physics, Mechanics, and Astronomy, 68, 100413

Huang, L., Cai, R.-G., & Wang, S.-J. 2025, Science China Physics, Mechanics, and Astronomy, 68, 100413

work page 2025
[23]

Kass, R. E. & Raftery, A. E. 1995, Journal of the american statistical association, 90, 773

work page 1995
[24]

2016, Reports on Progress in Physics, 79, 046902

Koyama, K. 2016, Reports on Progress in Physics, 79, 046902

work page 2016
[25]

2020, MNRAS, 491, 4960

Li, E.-K., Du, M., & Xu, L. 2020, MNRAS, 491, 4960

work page 2020
[26]

2023, Science China Physics, Mechanics, and Astronomy, 66, 229511

Li, S.-Y ., Li, Y .-L., Zhang, T., et al. 2023, Science China Physics, Mechanics, and Astronomy, 66, 229511

work page 2023
[27]

Linder, E. V . 2006, Phys. Rev. D, 74, 103518 Mc Conville, R. & Ó Colgáin, E. 2023, Phys. Rev. D, 108, 123533

work page 2006
[28]

D., Sáez-Chillón Gómez, D., & Sharov, G

Odintsov, S. D., Sáez-Chillón Gómez, D., & Sharov, G. S. 2025, European Phys- ical Journal C, 85, 298

work page 2025
[29]

& Skara, F

Perivolaropoulos, L. & Skara, F. 2022, New A Rev., 95, 101659

work page 2022
[30]

1999, ApJ, 517, 565

Perlmutter, S., Aldering, G., Goldhaber, G., et al. 1999, ApJ, 517, 565

work page 1999
[31]

Pierel, J. D. R., Coulter, D. A., Siebert, M. R., et al. 2025, ApJ, 981, L9

work page 2025
[32]

Pierel, J. D. R., Engesser, M., Coulter, D. A., et al. 2024, ApJ, 971, L32 Planck Collaboration, Aghanim, N., Akrami, Y ., et al. 2020, A&A, 641, A6

work page 2024
[33]

2024, MNRAS, 529, 2100

Popovic, B., Scolnic, D., Vincenzi, M., et al. 2024, MNRAS, 529, 2100

work page 2024
[34]

The Dark Energy Survey Supernova Program: A Reanalysis Of Cosmology Results And Evidence For Evolving Dark Energy With An Updated Type Ia Supernova Calibration

Popovic, B., Shah, P., Kenworthy, W. D., et al. 2025, arXiv e-prints, arXiv:2511.07517 Räsänen, S. 2006, J. Cosmology Astropart. Phys., 2006, 003

work page internal anchor Pith review Pith/arXiv arXiv 2025
[35]

G., Filippenko, A

Riess, A. G., Filippenko, A. V ., Challis, P., et al. 1998, AJ, 116, 1009

work page 1998
[36]

G., Yuan, W., Macri, L

Riess, A. G., Yuan, W., Macri, L. M., et al. 2022, ApJ, 934, L7

work page 2022
[37]

2011, ApJ, 738, 162 Sánchez, B

Sako, M., Bassett, B., Connolly, B., et al. 2011, ApJ, 738, 162 Sánchez, B. O., Brout, D., Vincenzi, M., et al. 2024, ApJ, 975, 5

work page 2011
[38]

Schwarz, D. J. & Weinhorst, B. 2007, A&A, 474, 717

work page 2007
[39]

1978, Annals of Statistics, 6, 461

Schwarz, G. 1978, Annals of Statistics, 6, 461

work page 1978
[40]

M., Jones, D

Scolnic, D. M., Jones, D. O., Rest, A., et al. 2018, ApJ, 859, 101

work page 2018
[41]

SN 2025ogs: A Spectroscopically-Normal Type Ia Supernova at z = 2 as a Benchmark for Redshift Evolution

Siebert, M. R., Pierel, J. D. R., Engesser, M., et al. 2025, arXiv e-prints, arXiv:2512.19783

work page internal anchor Pith review Pith/arXiv arXiv 2025
[42]

1978, Commun

Sugiura, N. 1978, Commun. Stat. Theory Methods, 7, 13

work page 1978
[43]

2025, MNRAS, 541, 2585

Vincenzi, M., Kessler, R., Shah, P., et al. 2025, MNRAS, 541, 2585

work page 2025
[44]

H., Mortonson, M

Weinberg, D. H., Mortonson, M. J., Eisenstein, D. J., et al. 2013, Phys. Rep., 530, 87

work page 2013
[45]

Wilks, S. S. 1938, Annals of Mathematical Statistics, 9, 60

work page 1938
[46]

Wiltshire, D. L. 2009, Phys. Rev. D, 80, 123512

work page 2009
[47]

2019, The Journal of Open Source Soft- ware, 4, 1298 Article number, page 11 of 11

Zonca, A., Singer, L., Lenz, D., et al. 2019, The Journal of Open Source Soft- ware, 4, 1298 Article number, page 11 of 11

work page 2019