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The Redshift-Distance Relation I Proc. Natl. Acad. Sci. USA Vol. 90, pp. 4798-4805, June 1993 Colloquium Paper This paper was presented at a colloquium entitled "Physical Cosmology," organized by a committee chaired by David N. Schramm, held March 27 and 28, 1992, at the National Academy of Sciences, Irvine, CA. The redshift-distance relation I. E. SEGAL Department of Mathematics 2-244, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139 ABSTRACT Key predictions of the Hubble law are incon- quintessential problem of observational astronomy, the in- sistent with direct observations on equitable complete samples herent cutoffon flux, produces a kind ofinherent bias, so that ofextragalactic sources in the optical, infrared, and x-ray wave "fair" here doesn't mean that the sources are from the same bands-e.g., the predicted dispersion in apparent magnitude is population at all redshifts but that the criterion for inclusion persistently greatly in excess of its observed value, precluding in the sample is an objective and theory-independent one. an explanation via hypothetical perturbations or irregularities. In this era of full disclosure, I would be remiss not to In contrast, the predictions of the Lundmark (homogeneous mention that I am the proposer ofa non-Doppler theory ofthe quadratic) law are consistent with the observations. The Lund- redshift, called chronometric cosmology (CC), that predicts mark law moreover predicts the deviations between Hubble law a different redshift-distance relation from those ofFriedman- predictions and observation with statistical consistency, while Lemaitre cosmology (FLC). This may affect our visceral the Hubble law provides no explanation for the close fit of the responses, but we have absolutely no interest in being Lundmark law. The flux-redshift law F X (1 + z)/z appears purveyors of an ultimately false theory of the nature of the consistent with observations on equitable complete samples in redshift, to speak also for my principal collaborator, Jeffrey the entire observed redshift range, when due account is taken Nicoll, as well as a number of younger colleagues. We have of flux limits by an optimal statistical method. Under the gone to considerable lengths to devise and implement pro- theoretical assumption that space is a fixed sphere, as in the cedures that are ofmaximal statistical efficiency and physical Einstein universe, this law implies the redshift-distance rela- conservatism; to document these procedures so that our tion z = tan2(r/2R), where R is the radius of the spherical results are fully reproducible from the observations in the space. This relation coincides with the prediction of chrono- designated sample; to base our analyses on well-documented metric cosmology, which estimates R as 160 ± 40 Mpc (1 parsec and thoroughly studied flux-limited samples (in whose ob- = 3.09 x 1016 m) from the proper motion to redshift relation servation we had no hand, and which are used exactly as of superluminal sources. Tangential aspects, including statis- published, without any type of model-dependent correction); tical methodology, fundamental physical theory, bright cluster and to stick to the same basic procedure in all the analyses, galaxy samples, and proposed luminosity evolution, are briefly in all redshift ranges and for all types of sources, to facilitate considered. fair comparisons. For conservatism on the physical side, we make no as- The good news is that there is a simple redshift relation that sumptions regarding the spatial distribution of the sources. appears consistent with observations in complete objectively These are uncertain, often questioned, and unnecessary for defined samples in the infrared and x-ray wave bands as well the estimation of the luminosity function (LF), when nor- as the optical. The bad news is that it doesn't at all resemble malized to a total ofunity, as suffices for cosmological testing the Hubble law, which appears simply irreconcilable with purposes. For conservatism on the observational side we these observations. make no assumption as to the completeness of the sample in These facts come out in a systematic audit of redshift redshift (e.g., there may be gaps in the redshift distribution). observations and theory, of the sort that every putative All that is assumed is that at each redshift that is observed, scientific theory should have periodically, as emphasized in there is no selection on the basis of flux, down to the the National Academy of Sciences booklet "Science and designated limit of the sample. The conservatism of this Creationism" (1). Outside of logic and mathematics, our assumption is further increased by analysis of successively basis for separating fact from fancy has to be probability, brighter subsamples. Indeed, the flux limits may vary with whose elementary laws are beyond dispute. A theory whose the object, as when different parts of the sky are observed predictions are improbably deviant from direct observation is down to different limits. scientifically incorrect, while one whose predictions agree Conservatism on the probabilistic side is attained by being with direct observation within apparent statistical fluctua- nonparametric, computer-intensive, and using maximum- tions is scientifically tenable and may be correct, although likelihood theory, all at the same time. At first glance, it may never in an ultimate sense. A "theory" that escapes devia- appear paradoxical to be both nonparametric and maximum- tions from direct observation by not making predictions just likelihood, because the latter theory has the goal of estimat- isn't a scientific theory, or in Pauli's terms, "isn't even good ing a finite set of parameters, in terms of which the unknown enough to be wrong." Thus the burden of our audit has to be the objective, Abbreviations: LF, luminosity function; CC, chronometric cosmol- reproducible, maximally accurate determination of probabil- ogy; FLC, Friedman-Lemaitre cosmology; LE, luminosity evolu- ity levels for the deviations of theoretical prediction from tion; IRAS, infrared astronomical satellite; SIRAS, galaxy sample observation, in documented fair samples. Of course, the treated in ref. 14; CMB, cosmic microwave background; EU, Ein- stein universe; SR, special relativity; MP, Mach's principle; EEP, Einstein equivalence principle; BQS, quasar sample treated in ref. 7; The publication costs of this article were defrayed in part by page charge SLF, sample LF estimated by ROBUST; QDOT, galaxy sample payment. This article must therefore be hereby marked "advertisement" treated in refs. 17 and 18; EMSS, x-ray source sample treated in refs. in accordance with 18 U.S.C. §1734 solely to indicate this fact. 19, 20, and 42; AGN, active galactic nucleus; GR, General Relativity. 4798 Downloaded by guest on October 1, 2021 Colloquium Paper: Segal Proc. Natl. Acad. Sci. USA 90 (1993) 4799 distribution function (here the LF, taken throughout in nor- observed and for artificial samples. On the basis of the latter malized form) is expressed. But, following a suggestion from trials, there is no question that the application of ROBUST to Michael Woodroofe, we simply divided the absolute magni- an artificial flux-limited sample constructed assuming any tude (or log flux) range subject to "Malmquist bias" into so given cosmology and LF will reproduce that LF within many bins and used the values of the differential LF, assumed statistical fluctuations, irrespective of the assumed given constant in each bin, as the parameters. A priori this appears redshifts. as a rather horrible nonlinear problem, involving extensive The results of parallel analyses of the low redshift to number crunching with no assurance of convergence. For- distance laws (e.g., for z < 0.1) predicted by FLC or CC tunately, we found a very simple closed-form solution, ex- (briefly, the Hubble law, or Cl, and the Lundmark law, or C2, pressible simply as a rational function of the occupation where Cp denotes the redshift-distance law of exponent p), numbers of the bins. Indeed, the solution was so natural that on the bases of large, clearly equitable, and objectively we guessed it to begin with and then determined, following a defined flux-limited samples, have been qualitatively similar. suggestion from Herman Chernoff, that it was indeed max- (Historical note: the Lundmark law is named primarily in imum-likelihood (2-3). We called our method ROBUST recognition of his methodological priority; the empirical law because it made no a priori assumption as to a parametric he published in 1925 was quadratic but distinct from C2.) The form for the LF. Cl predicted mean apparent magnitude at given redshifts is Donald Lynden-Bell then called his heuristic infinite- too bright on average, and quite conspicuously so at the number-of-parameters maximum-likelihood analysis (4) to lowest redshifts, which would be the least affected by as- our attention. While this work involved the assumption of sumed luminosity evolution (LE). Relatedly, the slope of the spatial uniformity of the source distribution, as ours did not, predicted magnitude-redshift relation consistently exceeds the resulting LF estimate was nevertheless the limit of the that observed. The C2 predicted mean apparent magnitudes ROBUST estimate as the bin size tends to zero. are considerably closer to the observed values, and the Maximum-likelihood theory, which for practical purposes deviations show no significant trend with redshift. goes back to Fisher (1922), requires a finite number of The correlation of absolute magnitude with (log) redshift is parameters and in these terms defines the notion of efficiency a highly relevant quantity because it speaks to the origin of (equivalent to that of maximal information content, roughly) the notion that redshift may be a function of distance, which and of sufficiency (extraction of all the information in the arises from the observed correlation of apparent magnitude sample relevant to the parameters being estimated). RO- with redshift. One of the most crucial functions of a cosmol- BUST goes beyond generic maximum-likelihood estimation ogy is to explain this correlation.
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