# Multiverse Predictions for Habitability: The Number of Stars and Their Properties

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Properties and Probabilities of Our Universe

#### 2.2. Drake Parameters

## 3. Number of Stars in the Universe ${N}_{\star}$

#### 3.1. What Is Meant by this Quantity?

#### 3.2. Is Habitability Simply Proportional to the Number of Stars?

## 4. Habitability Dependent on Stellar Properties

#### 4.1. Is Photosynthesis Necessary for Complex Life?

#### 4.2. Is Photosynthesis Possible around Red Dwarfs?

#### 4.3. Is There a Minimum Timescale for Developing Intelligence?

#### 4.4. Are Tidally Locked Planets Habitable?

#### 4.5. Are Convective Stars Habitable?

#### 4.6. Is Habitability Dependent on Entropy Production?

## 5. Discussion: Comparing 40 Hypotheses

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Stellar Properties

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1 | The code to compute all probabilities discussed in the text is made available at https://github.com/mccsandora/Multiverse-Habitability-Handler. |

2 | The above used the log-uniform prior for $\beta $ and $\gamma $, as discussed above. If instead we use a uniform prior, we find $\mathbb{P}\left({\alpha}_{\mathrm{obs}}\right)=0.20$, $\mathbb{P}\left({\beta}_{\mathrm{obs}}\right)=0.26$, and $\mathbb{P}\left({\gamma}_{\mathrm{obs}}\right)=3.1\times {10}^{-9}$. We see that the probabilities for $\alpha $ and $\beta $ are affected only slightly, and the probability for $\gamma $ is decreased by two orders of magnitude. This is a fairly typical result. |

3 | We entertain adopting a different stance as to whether planets must be in the temperate zone to be habitable in [29]. |

4 | The reader may object that even if a star’s planets become tidally locked before it expires, they may remain tidally unlocked for sufficient time for complex life to develop. We have adopted this more rough criterion to make this section self contained, but one may instead compare it to the biological time discussed in the previous subsection, for example. If this is done, they would find ${\lambda}_{\mathrm{TL}}=3556{\alpha}^{47/17}{\beta}^{12/17}{\gamma}^{-4/17}$, which does not alter the conclusions by much. |

**Figure 1.**The underlying logic behind this task. We consider separate habitability hypotheses ‘H’, and determine whether our universe is good at H, in the sense that adopting this notion of habitability makes our presence in this universe probable (and equally importantly, that not adopting this notion makes our presence in this universe improbable). This yields a prediction for whether or not H is a good requirement for habitability that can then be tested against upcoming observations. There are dozens of proposed habitability criteria in the literature, and though not all of them will have a significant influence on our likelihood, many will. If we do live in a multiverse, we expect compatibility with all of these tests; if just one yields incompatible results, we will be able to rule out the multiverse hypothesis to a potentially very high degree of confidence.

**Figure 2.**The distribution of stars throughout the multiverse. Each point represents a universe with a given set of parameters; the black dot represents our values. A strong preference for large $\gamma $, weak preference for small $\beta $, and a slightly stronger preference for small $\alpha $ value can be seen. Note that the $\gamma $ axes are logarithmic, as well as the color display for the probability, which spans 16 order of magnitude.

**Figure 3.**The fraction of stars which are capable of supporting photosynthesis as a function of the composite parameter Y defined in the text. The different curves correspond to taking the minimal wavelength to be both 400 nm and 600 nm, and the maximal to be 750 and 1100 nm. The solid curves use the estimate in Equation (8), and the dashed curves use the more refined initial mass function (IMF) of Equation (26).

**Figure 4.**Distribution of observers from imposing the photosynthesis condition. A strong preference for the parameters to be restricted to the photosynthetic range is introduced, but there is still a preference for large $\gamma $.

**Figure 5.**Distribution of observers from imposing the biological timescale condition. Of note is the secondary preference for large $\beta $ the anthropic boundary induces.

**Figure 8.**Habitable range of stellar masses for various choices of requirements. The shaded regions are treated as inhospitable for the various habitability assumptions.

**Table 1.**Dependence of the probabilities of our observed quantities on the upper and lower limits of the photosynthetic range.

Wavelength Range | $\mathbb{P}\left({\mathit{\alpha}}_{\mathbf{obs}}\right)$ | $\mathbb{P}\left({\mathit{\beta}}_{\mathbf{obs}}\right)$ | $\mathbb{P}\left({\mathit{\gamma}}_{\mathbf{obs}}\right)$ |
---|---|---|---|

400–1100 nm | 0.318 | 0.231 | 5.18e-07 |

400–750 nm | 0.242 | 0.263 | 3.85e-07 |

600–1100 nm | 0.444 | 0.175 | 7.35e-07 |

600–750 nm | 0.334 | 0.221 | 5.40e-07 |

**Table 2.**Probabilities of observing our values of parameters for various habitability hypotheses. Here the shorthands are photo: Photosynthesis criterion, TL: Tidal locking, conv: Convective stars, and bio: The biological timescale criterion.

Criteria | $\mathbb{P}\left({\mathit{\alpha}}_{\mathbf{obs}}\right)$ | $\mathbb{P}\left({\mathit{\beta}}_{\mathbf{obs}}\right)$ | $\mathbb{P}\left({\mathit{\gamma}}_{\mathbf{obs}}\right)$ |
---|---|---|---|

number of stars | 0.198 | 0.437 | 4.15 × 10${}^{-7}$ |

bio | 0.281 | 0.116 | 2.52 × 10${}^{-5}$ |

conv | 0.183 | 0.426 | 3.1 × 10${}^{-7}$ |

conv bio | 0.0564 | 0.159 | 2.59 × 10${}^{-5}$ |

TL | 0.152 | 0.37 | 2.34 × 10${}^{-7}$ |

TL bio | 0.0101 | 0.413 | 8.27 × 10${}^{-5}$ |

TL conv | 0.152 | 0.37 | 2.34 × 10${}^{-7}$ |

TL conv bio | 0.0101 | 0.413 | 8.27 × 10${}^{-5}$ |

photo | 0.439 | 0.183 | 8.16 × 10${}^{-7}$ |

photo bio | 0.0631 | 0.103 | 1.7 × 10${}^{-5}$ |

photo conv | 0.439 | 0.183 | 8.06 × 10${}^{-7}$ |

photo conv bio | 0.0637 | 0.104 | 1.7 × 10${}^{-5}$ |

photo TL | 0.48 | 0.232 | 6.88 × 10${}^{-7}$ |

photo TL bio | 0.0352 | 0.281 | 0.000139 |

photo TL conv | 0.48 | 0.232 | 6.88 × 10${}^{-7}$ |

photo TL conv bio | 0.0352 | 0.281 | 0.000139 |

yellow | 0.486 | 0.162 | 8.78 × 10${}^{-7}$ |

yellow bio | 0.0351 | 0.102 | 1.72 × 10${}^{-5}$ |

yellow conv | 0.486 | 0.162 | 8.78 × 10${}^{-7}$ |

yellow conv bio | 0.0351 | 0.102 | 1.72 × 10${}^{-5}$ |

yellow TL | 0.0303 | 0.0308 | 1.63 × 10${}^{-6}$ |

yellow TL bio | 0.324 | 0.335 | 0.0114 |

yellow TL conv | 0.0303 | 0.0308 | 1.63 × 10${}^{-6}$ |

yellow TL conv bio | 0.324 | 0.335 | 0.0114 |

**Table 3.**Probabilities of observing our values of parameters for various habitability hypotheses with entropy condition (denoted by S) included. The other shorthands are the same as above.

Criteria | $\mathbb{P}\left({\mathit{\alpha}}_{\mathbf{obs}}\right)$ | $\mathbb{P}\left({\mathit{\beta}}_{\mathbf{obs}}\right)$ | $\mathbb{P}\left({\mathit{\gamma}}_{\mathbf{obs}}\right)$ |
---|---|---|---|

photo S | 0.24 | 0.386 | 0.376 |

photo bio S | 0.178 | 0.414 | 0.426 |

photo conv S | 0.256 | 0.401 | 0.368 |

photo conv bio S | 0.191 | 0.433 | 0.421 |

photo TL S | 0.394 | 0.446 | 0.356 |

photo TL bio S | 0.278 | 0.465 | 0.453 |

photo TL conv S | 0.394 | 0.446 | 0.356 |

photo TL conv bio S | 0.278 | 0.465 | 0.453 |

yellow S | 0.191 | 0.45 | 0.317 |

yellow bio S | 0.125 | 0.486 | 0.38 |

yellow conv S | 0.191 | 0.45 | 0.317 |

yellow conv bio S | 0.125 | 0.486 | 0.38 |

yellow TL S | 0.481 | 0.396 | 0.44 |

yellow TL bio S | 0.343 | 0.476 | 0.31 |

yellow TL conv S | 0.481 | 0.396 | 0.44 |

yellow TL conv bio S | 0.343 | 0.476 | 0.31 |

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Sandora, M.
Multiverse Predictions for Habitability: The Number of Stars and Their Properties. *Universe* **2019**, *5*, 149.
https://doi.org/10.3390/universe5060149

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Sandora M.
Multiverse Predictions for Habitability: The Number of Stars and Their Properties. *Universe*. 2019; 5(6):149.
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2019. "Multiverse Predictions for Habitability: The Number of Stars and Their Properties" *Universe* 5, no. 6: 149.
https://doi.org/10.3390/universe5060149