ABSTRACT We present an extended analytical model for cosmic star formation, with the aim of investigating the impact of cosmological parameters on the star formation history within the $\Lambda$CDM paradigm. Constructing an ensemble of flat $\Lambda$CDM models where the cosmological constant varies between $\Lambda = 0$ and $10^5$ times the observed value, $\Lambda _{\rm obs}$, we find that the fraction of cosmic baryons that are converted into stars over the entire history of the universe peaks at $\sim$ 27 per cent for $0.01 \lesssim \Lambda /\Lambda _{\rm obs} \lesssim 1$. We explain, from first principles, that the decline of this asymptotic star formation efficiency for lower and higher values of $\Lambda$ is driven, respectively, by the astrophysics of star formation, and by the suppression of cosmic structure formation. However, the asymptotic efficiency declines slowly as $\Lambda$ increases, falling below 5 per cent only for $\Lambda \gt 100 \, \Lambda _{\rm obs}$. Making the minimal assumption that the probability of generating observers is proportional to this efficiency, and following Weinberg in adopting a flat prior on $\Lambda$, the median posterior value of $\Lambda$ is $539 \, \Lambda _{\rm obs}$. Furthermore, the probability of observing $\Lambda \le \Lambda _{\rm obs}$ is only 0.5 per cent. Although this work has not considered recollapsing models with $\Lambda \lt 0$, the indication is thus that $\Lambda _{\rm obs}$ appears to be unreasonably small compared to the predictions of the simplest multiverse ensemble. This poses a challenge for anthropic reasoning as a viable explanation for cosmic coincidences and the apparent fine-tuning of the Universe: either the approach is invalid or more parameters than $\Lambda$ alone must vary within the ensemble.
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