From the perspective of fundamental physics, the cosmological constant would be expected to be either of the order of the Planck density (i.e. enormous), or to be exactly zero due to some symmetry principle. The fact that astronomical evidence implies that the cosmological constant is small but non-zero is a puzzle. The fact that its magnitude happens to be comparable with the mean matter density of the universe in the current epoch is an additional puzzle.

Martel, Shapiro and Weinberg (1998) estimated the probability distribution for lambda under the anthropic constraint that it is not so large as to prevent structure formation. Their result was that Omega-lambda (i.e. lambda normalised by the critical density) would be expected to be greater than ~0.6. In the event, WMAP measurements favour a value for Omega-lambda of ~0.7, broadly consistent with the anthropic argument.

More recently, Weinberg (2005) has updated the calculations using the fluctuation measurements from WMAP. He finds that the anthropic argument yields a probability that Omega-lambda is ~0.7 of about 7% to 16%, depending upon the size scale used for the fluctuation smoothing (these results being based upon 1 to 2 Mpc respectively). These probabilities seem large enough to maintain the credibility of anthropic reasoning.

However, Martel et al assumed that lambda must be positive. Whilst this appears to be true in our universe, it is not valid to assume it when calculating the probability distribution for lambda. Sufficiently large and negative lambda will be anthropically vetoed, since such a universe will not live long enough for life to evolve. However, one might expect structure formation to be enhanced by a sufficiently small but negative lambda. Provided that such universes have sufficiently long lifetimes they might contribute equally, or even dominate, the population of universes within the multiverse which give rise to observers. If so, Weinberg’s estimate of 7% - 16% as the anthropic probability of our actual Omega-lambda (~0.7) may be substantially over-estimated. This would rebound unfavourably on anthropic reasoning.

In any case, Tegmark and Rees (1997) claim that there is a flaw in the reasoning of Martel et al. The formation of structure depends sensitively on the magnitude of the primordial matter fluctuations. Consequently, a change in lambda by a factor f can be accomodated without prejudice to structure formation provided that Q is also changed by a factor f^1/3. This is discussed further in CCC12B. Hence, the programm of Martel et al should really consider a bivariate probability distribution in both lambda and Q.

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A remakable picture of two galaxies in the process of merging (NGC 2623, or Arp243, in the Crab, 250 million light years distant and 50,000 light years across). Such galactic mergers can take hundreds of millions of years to complete. Picture from Hubble (optical) plus Spitzer (infrared), XMM-Newton (X-ray), and GALEX (Ultraviolet). [Credit: NASA, ESA and A. Evans (Stony Brook)]