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Distance measures in cosmology

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ReviewThe objections and verdict, as JSON Paper breakdownThe authors' claims, conditions and measurements, as JSON Signed receiptProof this review has not been altered
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David W. Hogg (IAS)

Astrophysics · Submitted 11 May 1999 · 10 references

Pith's verdict: The standard cosmography cheat sheet, formulae correct, with one real error in Eq. 10 where the (1+z) factor is inverted.

T0 · AI review

How this review was made: An AI referee (selfhosted/deepseek-v4.1-flash) read the paper on October 29, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

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  4. T3 Formal statement The claims written out in a formal math language a computer can read.
  5. T4 Proof Those claims proved, and the proof checked by computer.

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arXiv IDastro-ph/9905116 (version 4)
Pith IDODEWUTZE, Pith's permanent record of this paper and its review
CitationDavid W. Hogg (IAS). "Distance measures in cosmology." arXiv:astro-ph/9905116 (1999). https://pith.science/paper/ODEWUTZE
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The paper

The paper claims

Every standard cosmological distance measure is a rewrite of one line-of-sight integral of the expansion rate, so a single function of three density parameters generates them all.

The reading

This is a working reference sheet: it collects in one place the formulae that turn an observed redshift into a physical distance, a volume, a time, or a probability of intersecting an object, for a universe that is homogeneous, isotropic, and filled with matter, curvature, and a cosmological constant. Its organizing claim is that these measures are not independent constructions but restatements of one another. A single integral of $1/E(z)$ along the line of sight, with $E(z) = [�Omega_M(1+z)^3 + �Omega_k(1+z)^2 + �Omega_�Lambda]^{1/2}$, gives the line-of-sight comoving distance, and from it follow the transverse comoving distance (curvature turns the integral into a $�sinh$, a straight line, or a $�sin$), the angular diameter distance $D_A = D_M/(1+z)$, the luminosity distance $D_L = (1+z)D_M$, the distance modulus, the comoving volume element, the lookback time, and the probability that a line of sight hits an object. A reader who wants to know why very distant objects appear larger in angle than nearer ones, or how a survey converts a solid angle and a redshift slice into a volume, gets the answer here in closed form. The value is bookkeeping and terminology discipline as much as mathematics: the sheet insists that the transverse comoving distance not be called a proper distance, and that the angular diameter distance between two redshifts is not the difference of two individual angular diameter distances.

What carries the argument

The load-bearing object is $E(z) = [�Omega_M(1+z)^3 + �Omega_k(1+z)^2 + �Omega_�Lambda]^{1/2}$, the expansion rate at redshift $z$ in units of today's Hubble constant, together with its line-of-sight integral $D_C = D_H �int_0^z dz'/E(z')$. Because $dz/E(z)$ is proportional to the light travel time across a redshift interval divided by the scale factor at that time, this integral is simultaneously a comoving distance, a time, and a volume kernel. Everything else in the sheet is a change of variables away from it, which is what makes the collection a single argument rather than a list.

Extended reading notes

Core claim

The paper's central claim is that the standard cosmography of a homogeneous, isotropic universe is one function wearing many hats. Define $E(z)$ from the three density parameters as in Eq. 14; integrate $1/E(z)$ over redshift and multiply by the Hubble distance $D_H = c/H_0$; that single number, the line-of-sight comoving distance $D_C$, then yields every other measure by an algebraic step. Curvature enters only through the transverse step, Eq. 16, which is a $�sinh$, an identity, or a $�sin$ depending on the sign of $�Omega_k$; the angular diameter distance is $D_M/(1+z)$; the luminosity distance is $(1+z)D_M$; the distance modulus is $5�log_{10}(D_L/10�{pc})$ plus a k-correction that depends on the object's spectrum; the comoving volume element is a product of $D_A^2$, two powers of $(1+z)$, and $1/E(z)$; the lookback time is the same integral with an extra $1/(1+z)$; and the intersection probability is that same kernel multiplied by number density and cross section. The paper also gives an analytic closed form for the transverse comoving distance when $�Omega_�Lambda = 0$, and a two-redshift angular diameter distance for lensing that is explicitly not the difference of the two single-redshift values. The author flags one known defect in his own text: the lensing distance formula of Eq. 19 is stated to be incorrect for $�Omega_k < 0$, on the strength of a private communication.

Load-bearing premise

The whole reduction holds only if the universe is homogeneous and isotropic on the relevant scales and its expansion is governed by matter, curvature, and a cosmological constant alone, with radiation and every other component dropped from $E(z)$; the paper states this in Section 2 but never bounds the error that dropping radiation introduces at high redshift.

Editorial extensions

If this is right

  • Any distance, volume, age, or lensing geometry in a matter-plus-curvature-plus-lambda universe can be computed from three density parameters and a redshift, with no re-derivation.
  • The turnover of angular diameter distance near $z �sim 1$ is a generic prediction of the framework, so very distant objects are predicted to subtend larger angles than objects at intermediate redshift.
  • Luminosity distance and angular diameter distance are locked together by $D_L = (1+z)^2 D_A$, so any two independent measurements of the same source constrain the same geometry.
  • Number counts and absorption-line statistics share the factor $(1+z)^2/E(z)$, so a survey's volume element and its line-of-sight hit probability are two readings of one kernel.
  • A single analytic expression covers the flat and open cases for the total comoving volume out to a redshift, making all-sky volume estimates a one-line calculation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because every measure is a functional of the same $E(z)$, ratios of distances at a common redshift test the metric rather than the world model; a measured departure from $D_L = (1+z)^2 D_A$ would point to non-FRW geometry or to unaccounted absorption rather than to a different choice of $�Omega_M$ and $�Omega_�Lambda$.
  • The sheet drops radiation from $E(z)$. Anyone extending the same integral to the microwave background or to twenty-one-centimetre cosmology must add an $�Omega_R(1+z)^4$ term, which is a one-term edit but shifts the inferred sound horizon and hence every calibrated ruler.
  • The redshift at which angular diameter distance turns over is itself model dependent, so locating that turnover empirically is a cheap consistency check on the density parameters that requires no standard candle.
  • The two-redshift lensing distance, flagged as wrong for negative curvature, is a place where the sheet's own reduction chain has an unfinished link; a corrected expression would be the natural next entry in the collection.
Original abstract

Formulae for the line-of-sight and transverse comoving distances, proper motion distance, angular diameter distance, luminosity distance, k-correction, distance modulus, comoving volume, lookback time, age, and object intersection probability are all given, some with justifications. Some attempt is made to rationalize disparate terminologies, or at least abuse bad usage.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,327 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.30 · human participants 0.02 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.07 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v4. Results and the main claim are underlined; .

Formulae for the line-of-sight and transverse comoving distances, proper motion distance, angular diameter distance, luminosity distance, k-correction, distance modulus, comoving volume, lookback time, age, and object intersection probability are all given, some with justifications. Some attempt is made to rationalize disparate terminologies, or at least abuse bad usage.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Formulae for the line-of-sight and transverse comoving distances, proper motion distance, angular diameter distance, luminosity distance, k-correction, distance modulus, comoving volume, lookback time, age, and object intersection probability are all given, some with justifications. contribution 0.98 0.81 0.21 asserted 1.00
s1 Some attempt is made to rationalize disparate terminologies, or at least abuse bad usage. contribution 0.89 0.86 0.07 hedged possibility 0.51

What kind of paper the classifier reports this is: review 0.54 · evidence from proof or derivation 0.98.

What was examined, and what was not

sources read
  • arXiv listing abstract v4
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. The dimensionless proper motion distance
Figure 2. The dimensionless angular diameter distance
Figure 3. The dimensionless luminosity distance DL/DH. The three curves are for the three world models, (ΩM, ΩΛ) = (1, 0), solid; (0.05, 0), dotted; and (0.2, 0.8), dashed. 12
Figure 4. The distance modulus DM. The three curves are for the three world models, (ΩM, ΩΛ) = (1, 0), solid; (0.05, 0), dotted; and (0.2, 0.8), dashed. 13
Figure 5. The integral of the comoving volume element from th
Figure 6. The dimensionless lookback time tL/tH and age t/tH. Curves cross at the redshift at which the Universe is half its present age. The three curves are for the three world models, (ΩM, ΩΛ) = (1, 0), solid; (0.05, 0), dotted; and (0.2, 0.8), dashed. 15
Figure 7. The dimensionless differential intersection prob

Figures reproduced from arXiv:astro-ph/9905116 by the authors.

References
How later papers used it

268 later papers cite it

Evidence: substantial. 264 of the citing papers share no author with it, and 78 of those carry a confident label from Pith's citation classifier.

Pith read 38 of the 312 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

Uses its method or tool 22 of 73 papers with this label
How the literature uses it

260 arXiv papers cite this in 304 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 180 sentences without a confident label

Contests it 1 paper
Uses its method or tool 67 papers
Uses its data or values 5 papers
Compares with it 1 paper
Cites it as background 43 papers

Discussion

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