Hubble constant — method and limits

Sources and terms of use

No permission was sought for Huchra's list (the author died in 2010; who maintains the copy and holds the rights could not be established). Table A2 prints 164 rows while the text counts 163: the row "1998, 62 ± 6, Croft and Dailey" is printed twice and the second copy is dropped. The remainder, 120 Croft & Dailey + 8 (1976–1989) + 35 (2011–2019), matches the paper's own breakdown. 8 rows have different upper and lower errors; the printed form is kept in the err_as_printed column of the CSV.

Where the two compilations overlap (1976–2010)

Two rows are taken as the same estimate when their years are at most one apart, H0 differs by at most 0.5, and the errors differ by at most 0.5 or 10% of the larger one (for the named rows of 1976–1989 the surname must also appear in Huchra's reference text). Of the 128 Table A2 rows in this span, 69 pair with a Huchra row (42 with identical year, value and error) and 59 are absent from Huchra's list; 65 of the 120 Croft & Dailey rows pair. A merged series (paired rows counted once) appears only in the sensitivity tables, since pairs matched on values alone may remove too much. The row-level comparison of Huchra and Table A2 is not published.

What the errors mean

Neither compilation states the confidence level of its errors. The explanatory text of Huchra's list says that external errors are used when available and otherwise the errors quoted by the authors, appropriately rounded; the Table A2 paper uses the errors as σ_i in χ² and says that "the given error bars account for all statistical effects". Neither is a statement of a confidence level. The chapter therefore treats the errors as one "unstated" series and does not relabel them as 1σ. Asymmetric errors enter as the mean of the two. For the record, Chen, Gott & Ratra (2003) treated Huchra's upper and lower errors as one-standard-deviation error bars; that is the authors' reading, not a statement of the compilation.

Metrics and decisions

• H1 precision half-life. The yearly median of the relative error (u/|x|), log₂-regressed on year (main value); the precision frontier (the smallest relative error of each year) as a reference. Interval: t interval of the slope.
• H2, not produced. M2 of the physical-constants chapter compares the move between two consecutive editions with their uncertainties. The H0 analogue, a pair of consecutive estimates, is only a publication order: several estimates share a year, so how the pairs are joined is arbitrary, and in the stage-1 sensitivity tables the verdict moved between "above the normal expectation" and "overlapping" depending on whether pairs were pooled or grouped by type code. Pairing only the follow-ups of the same team would come closest to M2, but neither compilation has a column that identifies teams. The chapter therefore leaves H2 empty and publishes no number under it; the numbering stays so that H1, H3 and H4 visibly correspond to M1, M3 and M4.
• H3 surprise against a reference. z = (reference − x)/u_x, with the shares |z| > 1, |z| > 2, |z| < 0.674 (inside the interquartile band) and |z| > 2.326. Two references side by side: Planck 2018 (67.36 ± 0.54; Planck Collaboration, Planck 2018 results. VI (arXiv:1807.06209), Table 2, column TT,TE,EE+lowE+lensing, 68% limits. A value within the ΛCDM model.) and Riess et al. 2022 (73.04 ± 1.04; A. G. Riess et al. 2022 (SH0ES, arXiv:2112.04510), baseline result of the abstract (with systematics).). The reference's uncertainty does not enter z. Intervals: cluster bootstrap by publication year (10,000 resamples, seed 2026). For Huchra's list, intervals with the type code as cluster are shown as a sensitivity check (they are wider).
• H4 Student-t degrees of freedom. ν of a zero-centered Student-t fitted by maximum likelihood to the H3 z (2,000 bootstrap resamples).
• The 2018 row "67.4 ± 0.5" of Table A2. It is the rounded Planck 2018 value, which section 3 of the paper quotes as the value derived from the cosmic microwave background (the table's reference [96] is a standard-siren forecast paper). So that the reference itself does not enter the series, the row is left out of H3 and H4 against Planck (162 rows); against Riess all 163 rows are used.
• Start year. The baseline starts in 1929; values from 1950 are shown alongside. The two rows before 1929 in Huchra's list (1927 and 1928, both primary code O, both with an unsigned upper error) are left out of the baseline, as are the three rows without an error (1978, 1984, 1986).
• How aggregates are shown. The yearly and decade aggregates of Huchra's list give only counts and the quartiles (linear interpolation) of H0 and of the relative error, and a year with fewer than three rows gives its count only, since the median of one or two rows is a printed value. 15 of the 51 years from 1929 to 2010 are such years. The 1940s have no row.
• Numbering. The definitions differ from M1–M4, so the metrics are numbered H1–H4 (as A1–A5 in the atomic-weights chapter).

Method codes

Huchra's type codes are used as printed; no new classification is added. The meanings of the primary codes (method) and secondary codes (authors, conditions) are the file legend's own words, and codes absent from the legend (primary I and M, secondary T) are left unverified. 1 row has no code. Table A2 has no method column. Counts are over the rows with an error from 1929 on.

Primary codes

Secondary codes

Limits

• Neither compilation is complete. Even in the overlapping years nearly half of Table A2 is absent from Huchra's list, and most of Huchra's list is absent from Table A2.
• What an error means (statistical only or with systematics, and at what level) may differ from row to row; the compilations do not say.
• Several estimates share a year, so H1 rests on a yearly summary; many early years have a single point.
• Which side counts as "surprising" in H3 depends on the reference, which is why two are shown. The site does not decide whether the tension is real or which reference is right.
• Estimates sharing a calibrator, the same data or the same team are not independent; the year-cluster bootstrap handles only part of that dependence.
• The 1990–2010 rows of Table A2 cannot be traced to their papers, so the same paper may enter the two compilations once each.
• Table A2 was read from the arXiv version; in 1.2.1 all 164 rows were checked against the table of the published (MDPI web) version for year, value, error as printed and reference number, and all agree. The Planck 2018 value was checked in Results VI (arXiv 1807.06209v4), p. 16, Table 2; the Riess et al. 2022 value in arXiv 2112.04510v3, p. 1 (abstract) and p. 33.

Reproduction

Table A2 is read from the text layer of the arXiv PDF by h0_faerber.py into h0_faerber2020.csv. H1, H3 and H4 are computed by h0_stage1b.py, which calls the formulas of metrics.py through the adapter h0_metrics.py. The aggregates of Huchra's list come from h0_aggregate.py; the pages from h0_site.py (definitions in h0_registry.py). The original file of Huchra's list is not included in the public repository.