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2. The coincidences argument against Lambda

An argument against an observationally interesting value of Lambda, from our distrust of accidental coincidences, has been in the air for decades, and became very influential in the early 1980s with the introduction of the inflation scenario for the very early universe.

If the Einstein-de Sitter model in Eq. (35) were a good approximation at the present epoch, an observer measuring the mean mass density and Hubble's constant when the age of the universe was one tenth the present value, or at ten times the present age, would reach the same conclusion, that the Einstein-de Sitter model is a good approximation. That is, we would flourish at a time that is not special in the course of evolution of the universe. If on the other hand two or more of the terms in the expansion rate equation (11) made substantial contributions to the present value of the expansion rate, it would mean we are present at a special epoch, because each term in Eq. (11) varies with the expansion factor in a different way. To put this in more detail, we imagine that the physics of the very early universe, when the relativistic cosmological model became a good approximation, set the values of the cosmological parameters. The initial values of the contributions to the expansion rate equation had to have been very different from each other, and had to have been exceedingly specially fixed, to make two of the Omegai0's have comparable values. This would be a most remarkable and unlikely-looking coincidence. The multiple coincidences required for the near vanishing of adot and addot at a redshift not much larger than unity makes an even stronger case against Lemaître's coasting model, by this line of argument.

The earliest published comment we have found on this point is by Bondi (1960, p. 166), in the second edition of his book Cosmology. Bondi notes the "remarkable property" of the Einstein-de Sitter model: the dimensionless parameter we now call OmegaM is independent of the time at which it is computed (since it is unity). The coincidences argument follows and extends Bondi's comment. It is presented in McCrea (1971, p. 151). When Peebles was a postdoc, in the early 1960s, in R. H. Dicke's gravity research group, the coincidences argument was discussed, but published much later (Dicke, 1970, p. 62; Dicke and Peebles, 1979). We do not know its provenance in Dicke's group, whether from Bondi, McCrea, Dicke, or someone else. We would not be surprised to learn others had similar thoughts.

The coincidences argument is sensible but not a proof, of course. The discovery of the 3 K thermal cosmic microwave background radiation gave us a term in the expansion rate equation that is down from the dominant one by four orders of magnitude, not such a large factor by astronomical standards. This might be counted as a first step away from the argument. The evidence from the dynamics of galaxies that OmegaM0 is less than unity is another step (Peebles, 1984, p. 442; 1986). And yet another is the development of the evidence that the Lambda and dark matter terms differ by only a factor of three (Eq. [2]). This last is the most curious, but the community has come to accept it, for the most part. The precedent makes Lemaître's coasting model more socially acceptable.

A socially acceptable value of Lambda cannot be such as to make life impossible, of course. (17) But perhaps the most productive interpretation of the coincidences argument is that it demands a search for a more fundamental underlying model. This is discussed further in Sec. III.E and the Appendix.



17 If Lambda were negative and the magnitude too large there would not be enough time for the emergence of life like us. If Lambda were positive and too large the universe would expand too rapidly to allow galaxy formation. Our existence, which requires something resembling the Milky Way galaxy to contain and recycle heavy elements, thus provides an upper bound on the value of Lambda. Such anthropic considerations are discussed by Weinberg (1987, 2001), Vilenkin (2001), and references therein. Back.

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