The impact of the cosmological constant on past and future star formation

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.