Other pages in this series (6)

Precision half-life

Years for the relative error of published H0 estimates to halve

Takeaway

In Huchra's list (1929–2010) the yearly median relative error halved every 58.7 years (95% interval 38.0–128.5); from 1950 on, every 35.8 years.
Table A2 (1976–2019) gives 13.9 years (10.6–20.1). The precision frontier, the smallest error of each year, is shorter in every series.

Definition: For each year, the median of the relative error (u/|x|, u the quoted error) over that year's estimates; its log₂ is regressed on publication year (main value). The "precision frontier", the smallest relative error of each year, is shown as a reference. The two compilations are computed separately.
Formula: Half-life = −1 / b, b = slope of log₂(yearly median u/|x|) on year. 95% interval: t(n−2) interval of the slope. Asymmetric errors enter as the mean of the two.
Denominator: Huchra's list: 548 rows with an error from 1929 to 2010 over 51 years (543 rows, 47 years from 1950). Table A2: 163 rows from 1976 to 2019 over 37 years. The two rows of 1927 and 1928 are left out.
How to read it: The yearly-median half-life of Huchra's list is 58.7 years (95% interval 38.0–128.5), or 35.8 years (25.6–59.7) from 1950. Table A2 gives 13.9 years (10.6–20.1). The precision frontier is shorter in every series: 26.1, 17.7 and 11.8 years. The median over the core constants of the physical-constants chapter (M1) is 5.4 years.
Caveats: Several estimates share a year, so the metric rests on a yearly summary, and many early years have a single point (15 years have fewer than three rows). Neither compilation states the confidence level of its errors, so their meaning may differ from row to row. The two compilations differ in span and coverage, so their half-lives are not compared directly. The frontier is the smallest quoted error of the year, so an underestimated error shortens it as it stands.

Huchra’s list, aggregated by year: bar = quartile range of the relative error, dot = median. Years with fewer than three rows show only an × (count in the tooltip). Log vertical axis.

Table A2, one dot per row: relative error (quoted error ÷ H0, the mean if asymmetric). Log vertical axis.

Series	Start	Variant	Years (n)	Span	Slope log₂/year (SE)	Half-life (years)	95% interval	R²	Rows
Huchra's list	from 1929	Yearly median	51	1929–2010	−0.01705 (0.00461)	58.66	38.01–128.49	0.218	548
Huchra's list	from 1929	Precision frontier	51	1929–2010	−0.03827 (0.00671)	26.13	19.32–40.33	0.399	548
Huchra's list	from 1950	Yearly median	47	1951–2010	−0.02794 (0.00556)	35.79	25.55–59.73	0.360	543
Huchra's list	from 1950	Precision frontier	47	1951–2010	−0.05636 (0.00856)	17.74	13.59–25.56	0.491	543
Table A2	1976–2019	Yearly median	37	1976–2019	−0.07201 (0.01098)	13.89	10.60–20.12	0.551	163
Table A2	1976–2019	Precision frontier	37	1976–2019	−0.08471 (0.01223)	11.81	9.13–16.70	0.578	163
Merged (sensitivity)	from 1929	Yearly median	60	1929–2019	−0.02909 (0.00487)	34.37	25.74–51.72	0.381	642
Merged (sensitivity)	from 1929	Precision frontier	60	1929–2019	−0.04156 (0.00557)	24.06	18.97–32.88	0.490	642
Merged (sensitivity)	from 1950	Yearly median	56	1951–2019	−0.04231 (0.00555)	23.63	18.72–32.06	0.519	637
Merged (sensitivity)	from 1950	Precision frontier	56	1951–2019	−0.05536 (0.00676)	18.06	14.51–23.92	0.554	637

Download CSV: h0_h1_halflife.csv

Sources: J. Huchra, Estimates of the Hubble Constant (hubble.plot.dat, last updated 2010-10-07), Harvard-Smithsonian Center for Astrophysics; assembled as part of the NASA/HST Key Project on the Extragalactic Distance Scale. Aggregates only. · Faerber, T.; López-Corredoira, M. A Chi-Squared Analysis of the Measurements of Two Cosmological Parameters over Time. Universe 2020, 6, 114, doi:10.3390/universe6080114, Appendix Table A2 (read from arXiv:2008.04035v1). CC BY 4.0.